Peter G Ács

Mathieu Hoyrup

Crist Óbal Rojas

RANDOMNESS ON COMPUTABLE PROBABILITY SPACES-A DYNAMICAL POINT OF VIEW

Keywords: 3), Information Theory (H.1.1) Schnorr Randomness, Birkhoff's ergodic theorem, computable measures. PARTLY

We extend the notion of randomness (in the version introduced by Schnorr) to computable Probability Spaces and compare it to a dynamical notion of randomness: typicality. Roughly, a point is typical for some dynamic, if it follows the statistical behavior of the system (Birkhoff's pointwise ergodic theorem). We prove that a point is Schnorr random if and only if it is typical for every mixing computable dynamics. To prove the result we develop some tools for the theory of computable probability spaces (for example, morphisms) that are expected to have other applications.

Introduction

The roots of algorithmic randomness go back to the work of von Mises in the 20th century. He suggested a notion of individual infinite random sequence based on limit-frequency properties invariant under the action of selection functions from some "acceptable" set. The problem was then to properly define what an "acceptable" selection function could be. Some years later, the concept of computable function was formalized, providing a natural class of functions to be considered as acceptable. This gave rise to Church's notion of computable randomness. Nevertheless, substantial understanding was achieved only with the works of Kolmogorov [START_REF] Kolmogorov | Three approaches to the quantitative definition of information[END_REF], Martin-Löf [START_REF] Martin-Löf | The definition of random sequences[END_REF], Levin [START_REF] Zvonkin | The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms[END_REF] and Schnorr [START_REF] Schnorr | Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie[END_REF] and since then, many efforts have contributed to the development of this theory which is now well established and intensively studied.

There are several different possible definitions, but it is Martin-Löf's one which has received most attention. This notion can be defined, at least, from three different points of view:

(1) measure theoretic. This was the original presentation by Martin-Löf ( [START_REF] Martin-Löf | The definition of random sequences[END_REF]). Roughly, an infinite sequence is random if it satisfies all "effective" probabilistic laws (see definition 3.21).

(2) compressibility. This characterization of random sequences, due to Schnorr and Levin (see [START_REF] Zvonkin | The complexity of finite objects and the development of the concepts of information and randomness by means of the theory of algorithms[END_REF][START_REF] Schnorr | The process complexity and effective random tests[END_REF]), uses the prefix-free Kolmogorov complexity: random sequences are those which are maximally complex. (3) predictability. In this approach (started by Ville [START_REF] Ville | Etude Critique de la Notion de Collectif[END_REF] and reintroduced to the modern theory by Schnorr [START_REF] Schnorr | The process complexity and effective random tests[END_REF]) a sequence is random if, in a fair betting game, no "effective" strategy ("martingale") can win an unbounded amount of money against it. In [START_REF] Schnorr | Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie[END_REF], a somewhat broader notion of algorithmic randomness (narrower notion of probabilistic law) was proposed: Schnorr randomness. This notion received less attention over the years: Martin-Löfs definition is simpler, leads to universal tests, and many equivalent characterizations (besides, Schnorr's book is not in English. . . ). Recently, Schnorr randomness has begun to receive more attention. The work [START_REF] Downey | Schnorr randomness[END_REF] for instance, characterizes it in terms of Kolmogorov complexity.

In the present paper, first we extend Schnorr randomness to arbitrary computable probability spaces and develop some useful tools. Then, taking a dynamical systems point of view, we introduce yet another approach to the definition of randomness: typicality. Roughly, a point is typical for some measure-preserving ergodic dynamic, if it follows the statistical behavior of the system (given by Birkhoff's pointwise ergodic theorem) with respect to every bounded continous function used to follow its trajectory (or equivalently, every computable function, see Definition 3.28). We then show that:

Theorem. In any computable probability space, a point is Schnorr random if and only if it is typical for every mixing computable dynamical system.

The paper is organized as follows: Section 2 presents all needed concepts of computability theory and computable measure theory over general metric spaces. Parts of this section, for example on µ-computable functions, are new and should be of independent interest. Section 3.1 generalizes Schnorr randomness and studies some useful properties, after which we introduce the notion of typicality. Section 3.3 is devoted to the proof of our main result.

Computability

In classical recursion theory, a set of natural numbers is called recursively enumerable (r.e. for short) if it is the range of some partial recursive function. That is if there exists an algorithm listing (or enumerating) the set.

Strictly speaking, recursive functions only work on natural numbers, but this can be extended to the objects (thought of as "finite" objects) of any countable set, once a numbering of its elements has been chosen. We will sometimes use the word algorithm instead of recursive function when the inputs or outputs are interpreted as finite objects. The operative power of an algorithm on the objects of such a numbered set obviously depends on what can be effectively recovered from their numbers.

Examples 2.1. 1 N k can be numbered in such a way that the k-tuple of number i can be computed from i and vice versa. [START_REF] Downey | Schnorr randomness[END_REF] The set Q of rational numbers can be injectively numbered Q = {q 0 , q 1 , . . .} in an effective way: the number i of a rational a/b can be computed from a and b, and vice versa. We fix such a numbering.

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All through this work, we will use recursive functions over numbered sets to define computability or constructivity notions on infinite objects. Depending on the context, these notions will take particulars names (computable, recursively enumerable, r.e. open, decidable, etc...) but the definition will be always of the form: obect x is constructive if there exists a recursive ϕ: N → D satisfying property P(ϕ, x) (where D is some numbered set).

For example, E ⊂ N is r.e. if there exists a recursive ϕ:

N → N satisfying E = range(ϕ).
Each time, a uniform version will be implicitly defined: a sequence (x i ) i is constructive uniformly in i if there exists a recursive ϕ: N × N → D satisfying property P(ϕ(i, •), x i ) for all i.

In our example, a sequence (E i ) i is r.e. uniformly in i if there exists ϕ:

N × N → N satisfying E i = range(ϕ(i, •)) for all i.
Let us ilustrate this in the case of reals numbers (computable reals numbers were introduced by Turing in [START_REF] Turing | On computable numbers, with an application to the Entscheidungsproblem[END_REF]). Definition 2.2. A real number x ∈ R is said to be computable if there exists a total recursive ϕ :

N → Q satisfying |x -ϕ(n)| < 2 -n for all n ∈ N.
Hence by a sequence of reals (x i ) i computable uniformly in i we mean that there exists a recursive ϕ :

N × N → Q satisfying |x -ϕ(i, n)| < 2 -n for all n ∈ N, for all i ∈ N.
We also have the following notions:

Definition 2.3. Let x be a real number. We say that:

• x is lower semi-computable if the set {i ∈ N : q i < x} is r.e.,

• x is upper semi-computable if the set {i ∈ N : q i > x} is r.e.,

It is easy to see that a real number is computable if and only if it is lower and upper semi-computable.

Computable metric spaces

We breifly recall the basic of computable metric spaces. Definition 2.4. A computable metric space (CMS) is a triple X = (X, d, S), where

• (X, d) is a separable complete metric space.

• S = (s i ) i∈N is a numbered dense subset of X (called ideal points).

• The real numbers (d(s i , s j )) i,j are all computable, uniformly in i, j.

Some important examples of computable metric spaces:

Examples 2.5.

1 The Cantor space (Σ N , d, S) with Σ a finite alphabet. If x = x 1 x 2 . . . , y = y 1 y 2 . . . , are elements then the distance is defined by d(x, y) = i:x i =y i 2 -i . Let us fix some element of Σ denoting it by 0. The dense set S is the set of ultimately 0-stationary sequences. 2 (R n , d R n , Q n ) with the Euclidean metric and the standard numbering of Q n .

For further examples we refer to [START_REF] Weihrauch | Computability on computable metric spaces[END_REF]. The numbered set of ideal points (s i ) i induces the numbered set of ideal balls B := {B(s i , q j ) : s i ∈ S, q j ∈ Q >0 }. We denote by B i,j (or just B n ) the ideal ball B(s i , q j ), where •, • is a computable bijection between tuples and integers. 2 U i ∪ U j and U i ∩ U j are r.e. open uniformly in (i, j). See [START_REF] Hoyrup | Computability of probability measures and Martin-Löf randomness over metric spaces[END_REF].

Definition 2.6 (Computable points). A point x ∈ X is said to be computable if the set E x := {i ∈ N : x ∈ B i } is r.e.
Let (X, S X , d X ) and (Y, S Y , d Y ) be computable metric spaces. Let (B Y i ) i be the collec- tion of ideal balls from Y . Definition 2.9 (Computable Functions). A function T : X → Y is said to be computable if T -1 (B Y i ) is r.e.
open uniformly in i. It follows that computable functions are continuous. Since we will work with functions which are not necessarily continuous everywhere (and hence not computable), we shall consider functions which are computable on some subset of X. More precisely, a function

T is said to be computable on D (D ⊂ X) if T -1 (B Y i ) is r.e. open in D, uniformly in i. The set D is called the domain of computability of T .

Computable Probability Spaces

Let us recall some basic concepts of measure theory. Let X be a set. A family B of subsets of X is called an algebra

if (i)X ∈ B, (ii)A ∈ B ⇒ A C ∈ B and (iii) A, B ∈ B ⇒ A ∪ B ∈ B. We say that B is a σ-algebra if moreover A i ∈ B, i ≥ 1 ⇒ i A i ∈ B. If B 0
is a family of subsets of X, the σ-algebra generated by B 0 (denoted σ(B 0 )) is defined to be the smallest σ-algebra over X that contains B 0 . If B is a σ-algebra of subsets of X, we say that µ : B → [0, 1] is a probability measure if, for every family (A i ) i ⊂ B of disjoint subsets of X, the following holds:

µ( i A i ) = i µ(A i ). (3.1)
If X is a topological space, the Borel σ-algebra of X is defined as the σ-algebra generated by the family of open sets of X. Sets in the Borel σ-algebra are called Borel sets. In this paper, a probability space will always refer to the triple (X, B, µ), where B is the Borel σ-algebra of X and µ is a probability measure. A set A ⊂ X has measure zero if there is a Borel set A 1 such that A ⊂ A 1 and µ(A 1 ) = 0. We call two sets A 1 , A 2 ⊂ X equivalent modulo zero, and write A 1 = A 2 (mod 0), if the symmetric difference has measure zero. We write

A 1 ⊂ A 2 (mod 0) if A 1 is a subset of A 2 and A 1 = A 2 (mod 0).
When X is a computable metric space, the space of probability measures over X, denoted by M(X), can be endowed with a structure of computable metric space. Then a computable measure can be defined as a computable point in M(X).

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This is the distribution on the set of infinite binary sequences obtained by tossing a fair coin, and condition (3.1) simplifies to

λ(x0) + λ(x1) = λ(x).
Let X = (X, d, S) be a computable metric space. Let us consider the space M(X) of measures over X endowed with weak topology, that is:

µ n → µ iff µ n f → µf for all real continuous bounded f,
where µf stands for f dµ.

If X is separable and complete, then M(X) is separable and complete. Let D ⊂ M(X) be the set of those probability measures that are concentrated in finitely many points of S and assign rational values to them. It can be shown that this is a dense subset ( [START_REF] Billingsley | Convergence of Probability Measures[END_REF]).

We consider the Prokhorov metric ρ on M(X) defined by:

ρ(µ, ν) := inf{ǫ ∈ R + : µ(A) ≤ ν(A ǫ ) + ǫ for every Borel set A} where A ǫ = {x : d(x, A) < ǫ}.
This metric induces the weak topology on M(X). Furthermore, it can be shown that the triple (M(X), D, ρ) is a computable metric space (see [START_REF] Gács | Uniform test of algorithmic randomness over a general space[END_REF], [START_REF] Hoyrup | Computability of probability measures and Martin-Löf randomness over metric spaces[END_REF]).

Definition 3.2. A measure µ is computable if it is a computable point of (M(X), D, ρ)
The following result (see [START_REF] Hoyrup | Computability of probability measures and Martin-Löf randomness over metric spaces[END_REF]) will be intensively used in the sequel: Lemma 3.3. A probability measure µ is computable if and only if the measure of finite union of ideal balls µ(B i 1 ∪ . . . ∪ B i k ) is lower semi-computable, uniformly in i 1 , . . . , i k . Definition 3.4. A computable probability space (CPS) is a pair (X , µ) where X is a computable metric space and µ is a computable Borel probability measure on X.

As already said, a computable function defined on the whole space is necessarily continuous. But a transformation or an observable need not be continuous at every point, as many interesting examples prove (piecewise-defined transformations, characteristic functions of measurable sets,. . . ), so the requirement of being computable everywhere is too strong. In a measure-theoretical setting, the natural weaker condition is to require the function to be computable almost everywhere. In the computable setting this is not enough, and a computable condition on the set on which the function is computable is needed: Remark 3.8. Given a uniform sequence of µ-computable functions (f i ) i , any computable operation ⊙ n i=0 f i (adition, multiplication, composition, etc...) is µ-computable, uniformly in n.

Definition 3.5 (Constructive G δ -sets). We say that the set D ⊂ X is a constructive G δ -set if it
We recall that F : (X , µ) → (Y, ν) is measure-preserving if µ(F -1 (A)) = ν(A) for all Borel sets A. Definition 3.9 (morphisms of CPS's). A morphism of CPS's F : (X , µ) → (Y, ν), is a µ-computable measure-preserving function F :

D F ⊆ X → Y . An isomorphism of CPS's (F, G) : (X , µ) ⇄ (Y, ν) is a pair (F, G) of morphisms such that G • F = id on F -1 (D G ) and F • G = id on G -1 (D F ).
D ϕ = F -1 (G -1 (D F )) and D ϕ -1 = G -1 (D F ). Of course, (ϕ, ϕ -1
) is an isomorphism of CPS's as well.

The next proposition is a direct consequence of theorem 5.1.1 from [5]: Proposition 3.12. Every computable probability space is isomorphic to the Cantor space with an appropiate computable measure.

Definition 3.13. A set A ⊂ X is said to be almost decidable if the function 1 A : X → {0, 1} is µ-computable.
It is easy to see that a set A is almost decidable iff there is a constructive G δ set D of measure one and two r.e. open sets U and V such that:

U ∩ D ⊂ A, V ∩ D ⊆ A C , µ(U ) + µ(V ) = 1.
Remarks 3.14. 1 The collection of almost decidable sets is an algebra. 2 An almost decidable set is always a continuity set. 3 Ideal balls with zero boundary measure are always almost decidable. 4 Unless the space is disconnected (i.e. has non-trivial clopen subsets), no set can be decidable, i.e. semi-decidable (r.e.) and with a semi-decidable complement (such a set must be clopen 1 ). Instead, a set can be decidable with probability 1: there is an algorithm which decides if a point belongs to the set or not, for almost every point. This is why we call it almost decidable.

Ignoring computability, the existence of open sets with zero boundary measure directly follows from the fact that the collection of open sets is uncountable and µ is finite. The problem in the computable setting is that there are only countable many open r.e. sets. Fortunately, there still always exists a basis of almost decidables balls. Lemma 3.15. Let X be R or R + or [0, 1]. Let µ be a computable probability measure on X. Then there is a sequence of uniformly computable reals (x n ) n which is dense in X and such that µ({x n }) = 0 for all n.

Proof. Let I be a closed rational interval. We construct x ∈ I such that µ({x}) = 0. To do this, we construct inductively a nested sequence of closed intervals J k of measure < 2 -k+1 , with J 0 = I. Suppose J k = [a, b] has been constructed, with µ(J k ) < 2 -k+1 . Let m = (b -a)/3: one of the intervals [a, a + m] and [b -m, b] must have measure < 2 -k , and since their measure is upper-computable, we can find it effectively-let it be J k+1 .

From a constructive enumeration (I n ) n of all the dyadic intervals, we can construct x n ∈ I n uniformly. Corollary 3.16. Let (X , µ) be a CPS and (f i ) i be a sequence of uniformly computable real valued functions on X. Then there is a sequence of uniformly computable reals (x n ) n which is dense in R and such that µ({f -1 i (x n )}) = 0 for all i, n. Proof. Consider the uniformly computable measures µ i = µ • f -1 i and define ν = i 2 -i µ i . By Lemma 3.3, ν is a computable measure and then, by Lemma 3.15, there is a sequence of uniformly computable reals (x n ) n which is dense in R and such that ν({x n }) = 0 for all n. Since ν(A) = 0 iff µ i (A) = 0 for all i, we get µ({f -1 i (x n )}) = 0 for all i, n. The following result will be used many times in the sequel.

Corollary 3.17. There is a sequence of uniformly computable reals

(r n ) n∈N such that (B(s i , r n )) i,n is a basis of almost decidable balls. Proof. Apply Corollary 3.16 to (f i ) i defined by f i (x) = d(s i , x).
We remark that every ideal ball can be expressed as a r.e. union of almost decidable balls, and vice-versa. So the two bases are constructively equivalent. Definition 3.18. A computable probability space is a computable Lebesgue space if it is isomorphic to the computable probability space ([0, 1], m) where m is the Lebesgue measure.

Theorem 3.19. Every computable probability space with no atoms is a computable Lebesgue space.

Proof. We first prove the result for I = ([0, 1], µ).

Lemma 3.20. The interval endowed with a non-atomic computable probability measure is a computable Lebesgue space.

Proof. We define the morphism of the CPS as F (x) = µ([0, x]). As µ has no atom and is computable, F is computable and surjective. As F is surjective, it has right inverses. Two of them are G < (y) = sup{x : F (x) < y} and G > (y) = inf{x : F (x) > y}, and satisfy

F -1 ({y}) = [G < (y), G > (y)].
They are increasing and respectively left-and right-continuous. As F is computable, they are even lower-and upper semi-computable respectively. Let us define D = {y : G < (y) = G > (y)}: every y ∈ D has a unique pre-image by F , which is then injective on F -1 (D). The restriction of F on F -1 (D) has a left-inverse, which is given by the restriction of G < and G > on D. Let us call it G : D → I. By lower and upper semi-computability of G < and G > , G is computable. Now, D is a constructive G δ -set: D = n {y : G > (y) -G < (y) < 1/n}. We show that I \ D is a countable set. The family {[G < (y), G > (y)] : y ∈ I} indexed by I is a family of disjoint closed intervals, included in [0, 1]. Hence, only countably many of them have positive length. Those intervals correspond to points y belonging to I \ D, which is then countable. It follows that D has Lebesgue measure one (it is even dense). (F, G) is then an isomorphism between (I, µ) and (I, m). Now, we know from Theorem 3.12 that every CPS (X , µ) has a binary representation, which is in particular an isomorphism with the Cantor space (B N , µ ′ ). As mentioned in Example 3.10, the latter is isomorphic to (I, µ I ) where µ I is the induced measure. If µ is non-atomic, so is µ I . By the previous lemma, (I, µ I ) is isomorphic to (I, m).

Randomness and typicality

3.1.1. Algorithmic randomness. Definition 3.21. A Martin-Löf test (M L-test) is an uniform sequence (A n ) n of r.e. open sets such that µ(A n ) ≤ 2 -n . We say that x fails the M L-test if x ∈ A n for all n. A point x is called ML-random if it fails no M L-test. Definition 3.22. A Borel-cantelli test (BC-test) is a uniform sequence (C n ) n of r.e. open sets such that n µ(C n ) < ∞. We say that x fails the BC-test if x ∈ C n infinitly often (i.o.).
It is easy to show that: Proposition 3.23. x fails a M L-test iff x fails a BC-test. Definition 3.24. A Schnorr test (Sch-test) is a ML-test (A n ) n such that the sequence of reals (µ(A n )) n is uniformly computable. We say that x fails the Sch-test if x ∈ A n for all n. A point x is called Sch-random if it fails no Sch-test.

Definition 3.25. A strong BC-test is a BC-test (C n ) n such that n µ(C n ) is computable.
: x ∈ C n }| ≥ 2 k+c }. Then µ(A k ) < 2 -k . Observe that A k is the union of all the (2 k+c )-intersections of C n 's. Since µ(C k ) = n µ(C n ) -n =k µ(C n )
and the C n 's are r.e. we have that µ(C n ) is computable (uniformly in n). We choose a basis (B i ) i of almost decidable balls to work with. Recall that finite unions or intersections of almost decidable sets are almost decidable too and that the measure of an almost decidable set is computable. Now we show that µ(A k ) is computable uniformly in k. Let ǫ > 0 be rational. Let n 0 be such that

n≥n 0 µ(C n ) < ǫ 2 . Then µ( n≥n 0 C n ) < ǫ 2 . For each C n with n < n 0 we construct an almost decidable set C ǫ n ⊂ C n (a finite union of almost decidable balls) such that µ(C n ) -µ(C ǫ n ) < 1 n 0 ǫ 2 . Then n<n 0 [µ(C n ) -µ(C ǫ n ] < ǫ 2 .
Define A ǫ k to be the union of the (2 k+c )-intersections of the C ǫ n 's for n < n 0 . Then A ǫ k is almost decidable and then has a computable measure. Moreover

A k ⊂ A ǫ k ∪ ( n≥n 0 C n ) ∪ ( n<n 0 C n \ C ǫ n ), hence µ(A k ) -µ(A ǫ k ) < ǫ.
The following result is an easy modification of a result from [START_REF] Hoyrup | Computability of probability measures and Martin-Löf randomness over metric spaces[END_REF], so we omit the proof. Proposition 3.27. Morphisms of computable probability spaces are defined (and computable) on Schnorr random points and preserve Sch-randomness.

Dynamical systems and typicality

Let X be a metric space, let T : X → X be a Borel map. Let µ be an invariant Borel measure on X, that is:

µ(A) = µ(T -1 (A)) holds for each measurable set A. A set A is called T -invariant if T -1 (A) =
A modulo a set of measure 0. The system (T, µ) is said to be ergodic if each T -invariant set has total or null measure. In such systems the famous Birkhoff ergodic theorem says that time averages computed along µ-typical orbits coincide with space averages with respect to µ. More precisely, for any f ∈ L 1 (X) it holds

lim n→∞ S f n (x) n = f dµ, (3.2)
for µ-almost each x, where

S f n = f + f • T + . . . + f • T n-1 . If a point x satisfies equation (3.
2) for a certain f , then we say that x is typical with respect to the observable f . Definition 3.28. If x is typical w.r. to every bounded continuous function f : X → R, then we call it a T -typical point.

Remark 3.29. The proof of our main theorem will show as a side result that the definition would not change if we replaced "continuous" with "computable" in it.

In [START_REF] V'yugin | Effective convergence in probability and an ergodic theorem for individual random sequences[END_REF] is proved that ML-random infinite binary sequences are typical w.r. to any computable f . In [START_REF] Galatolo | Effective symbolic dynamics, random points, statistical behavior, complexity and entropy[END_REF], this is generalized via effective symbolic dynamics to computable probability spaces and µ-computable observables.

To have the result for Sch-random points it seems that a certain "mixing" property or "loss of memory" of the system has to be required. This is naturally expressed by means of the correlation functions. For measureable functions f, g let

C(f, g) = µ(f • g) -µf • µg, C n (f, g) = C(f • T n , g).

For events A, B with indicator functions 1

A , 1 B let C n (A, B) = C n (1 A , 1 B ),
which measures the dependence between the events A and B at times n ≫ 1 and 0 respectively. Note that C n (A, B) = 0 corresponds, in probabilistic terms, to T -n (A) and B being independent events.

Let us say that a family of Borel sets E is essential, if for every open set U there is a sequence (E i ) i of borel sets in E such that ∪ i E i ⊂ U (mod 0) (see Section 3). Definition 3.30. We say that a system (X, T, µ) is (polynomially) mixing if there is α > 0 and an essential family E = {E 1 , E 2 , ...} of almost decidable events such that for each i, j there is c i,j > 0 computable in i, j such that

|C n (E i , E j )| ≤ c i,j n α for all n ≥ 1.
We say that the system is independent if all correlation functions C n (E i , E j ) are 0 for sufficiently large n.

Examples of non-mixing but still ergodic systems are given for instance by irrational circle rotations with the Lebesgue measure. Examples of mixing but not independent sytems are given by piecewise expandings maps or uniformly hyperbolic systems which have a

We are now able to prove that (i) ⇒ (ii). Let E ∈ E. Put f = 1 E . Observe that f is µ-computable. For δ > 0, define the deviation sets:

A f n (δ) = x ∈ X : S f n (x) n -f dµ > δ .
By Corollary 3.16 we can choose δ such that A f n (δ) is almost decidable. Then their measures are computable, uniformly in n.

By the Chebychev inequality,

µ(A f n (δ)) ≤ 1 δ 2 S f n (x) n -f dµ 2 L 2 .
Let us change f by adding a constant to have f dµ = 0. This does not change the above quantity. Then, by invariance of µ we have

S f n (x) n -f dµ 2 L 2 = S f n (x) n 2 dµ = 1 n 2 nf 2 dµ + 2 n 2 i<j<n f • T j-i f dµ and hence δ 2 µ(A f n (δ)) ≤ f 2 L 2 n + 2 n k<n |C k (f, f )| ≤ f 2 L 2 n + 2c f,f (1 -α)n α .
(Observe that α can be replaced by any smaller positive number, so we assume α < 1.) Hence, µ(A f n (δ)) ≤ Cn -α for some constant C. Now, it is easy to find a sequence (n i ) i∈N such that the subsequence (n -α i ) i is effectively summable and n i n i+1 → 1 (take for instance n i = i β with αβ > 1). This shows that the sequence A f n i (δ) is a strong BC-test. Therefore, if x is Sch-random then x belongs to only finitely many A f n i (δ) for any δ and hence the subsequence then we have:

S f n i n i -2(1 -β i )M ≤ S f n n ≤ S f n i+1 n i+1 + 2(1 -β i )M,
where M is a bound of f . To see this, for any k, l, β with β ≤ k/l ≤ 1:

S f k k - S f l l = 1 - k l S f k k - S f l-k • T l-k l ≤ (1 -β)M + (l -k)M l = 2(1 -β)M.
Taking β = β i and k = n i , l = n first and then k = n, l = n i+1 gives the result. Thus, we have proved that a Schnorr random point x satisfies equation (3.3) for any E ∈ E. Lemma 3.33 allows to conclude. The (ii) ⇒ (iii) part follows since any independent dynamic is in particular mixing.

To prove the (iii) ⇒ (i) part we will need the following proposition which is a strengthening of a result of Schnorr in [START_REF] Schnorr | Zufälligkeit und Wahrscheinlichkeit. Eine algorithmische Begründung der Wahrscheinlichkeitstheorie[END_REF]. The proof is somewhat technical, for lack of space we do not included here (see appendix). Proposition 3.34. If the infinite binary string ω ∈ (B N , λ) is not Schnorr random (w.r. to the uniform measure), then there exists an isomorphism Φ : (B N , λ) → (B N , λ) such that Φ(ω) is not typical for the shift transformation σ. Now we are able to finish the proof of our main result: suppose that x is not Schnorr random. We construct a dynamic T for which x is not T -typical. From Proposition 3.12 and Theorem 3.19 we know that there is an isomorphism η : (X , µ) → (B N , λ) (here, λ denotes the uniform measure). If x / ∈ dom(η), we can take any independent endomorphism and modify it in order to be the identity on x. It is cleary still an independent endomorphism (maybe with a smaller domain of computability) and x, being a fixed point, can't be Ttypical. So let x ∈ dom(η). Then η(x) is not Schnorr random in (B N , λ), since η as well as its inverse preserve Schnorr randomness. Then, by Proposition 3.34, Φ(η(x)) is not σ-typical, where σ is the shift which is clearly independent (cylinders being the essential events). Put ψ = Φ • η. Define the dynamics T on X by T = ψ -1 • σ • ψ. It is easy to see that T is independent for events of the form E = ψ -1 [w]. Since {ψ -1 [w] : w ∈ 2 * } form an essential family of almost decidable events, T is independent too. As ψ(x) is not σ-typical, x is not T -typical either.

Definition 2 . 7

 27 (R.e. open sets). We say that the set U ⊂ X is r.e. open if there is some r.e. set E ⊂ N such that U = i∈E B i . If U is r.e. open and D ⊂ X is an arbitrary set then the set A := U ∩ D is called r.e. open in D. Examples 2.8. 1 If the sequence (U n ) n is r.e. open uniformly in n, then the union n U n is an r.e. open set.

Example 3 . 1 (

 31 Measure over a Cantor space). As a special example, we can set X = B N where B = {0, 1} and λ([x]) = 2 -|x| , where |x| is the length of the binary string x ∈ {0, 1} * .

Example 3 . 7 .

 37 is the intersection of a sequence of uniformly r.e. open sets. Definition 3.6 (µ-computable functions). Let (X , µ) and Y be a CPS and a CMS respectively. A function f : (X , µ) → Y is µ-computable if it is computable on a constructive G δ -set (denoted as domf or D f ) of measure one. Let m be the Lebesgue measure on [0, 1]. The binary expansion of reals defines a function from non-dyadic numbers to infinite binary sequences which induces a m-computable function from ([0, 1], m) to {0, 1} N .

Example 3 . 10 .

 310 Let (B N , λ) the probability space introduced in Example 3.1 with the cointossing distribution λ over the infinite sequences. The binary expansion (see example 3.7) creates an isomorphism of CPS's between the spaces ([0, 1], m) and (B N , λ). Remark 3.11. To every isomorphism of CPS's (F, G) one can associate the canonical invertible morphism of CPS's ϕ = F | Dϕ with ϕ -1 = G| D ϕ -1 , where

Proposition 3 .

 3 26. An element x fails a Sch-test if and only if x fails a strong BC-test. Proof. Let (C n ) n be a strong BC-test. Let c be such that 2 c > n µ(C n ). Define the r.e. open set A k := {x : |{n

  to f dµ = µ(E). To show that for such points the whole sequence S f n (x) n converges to f dµ = µ(E), observe that if n i ≤ n < n i+1 and β i := n i n i+1

In the Cantor space for example (which is totally disconnected), every cylinder (ball) is a decidable set.Indeed, to decide if some infinite sequence belongs to some cylinder it suffices to compare the finite word defining the cylinder to the corresponding finite prefix of the infinite sequence.

This work is licensed under the Creative Commons Attribution-NoDerivs License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nd/3.0/.

distinguished ergodic measure (called SRB measure and which is "physical" in some sense) with respect to which the correlations decay exponentially (see [START_REF] Viana | Stochastic dynamics of deterministic systems[END_REF]). An example of a mixing system for which the decrease of correlations is only polynomial and not exponential, is given by the class of Manneville-Pomeau type maps (non uniformly expanding with an indifferent fixed point, see [START_REF] Isola | On systems with finite ergodic degree[END_REF]). For a survey see [START_REF] Young | What are SRB measures, and which dynamical systems have them?[END_REF].

Proof of the main result

Now we prove our main theorem. Theorem 3.31. Let (X , µ) be a computable probability space with no atoms The following properties of a point x ∈ X are equivalent.

(i) x is Schnorr random.

(ii) x is T -typical for every mixing endomorphism T .

(iii) x is T -typical for every independent endomorphism T .

Remark 3.32. If the measure µ is atomic, it is easy to see that:

(1) (X , µ) admits a mixing endomorphism if and only if µ = δ x for some x. In this case the theorem still holds, the only random point being x. (2) (X , µ) admits an ergodic endomorphism if and only if µ = 1 n (δ x 1 + ... + δ xn ) (where x i = x j , for all i = j). In this case, a point x is Schnorr random if and only if it is typical for every ergodic endomorphism if and only if it is an atom.

Proof. Let us first prove a useful lemma. Let E ⊂ X be a Borel set. Denote by 1 E its indicator function. The ergodic theorem says that the following equality holds for almost every point:

(3.3) Lemma 3.33. Let E be an essential family of events. If x satisfies equation (3.3) for all E ∈ E then x is a T -typical point.

Proof. We have to show that equation (3.3) holds for any bounded continuous observable f . First, we extend equation (3.3) to every continuity open set C.

Applying the same argument to X \ C gives the result.

Now we extend the result to bounded continuous functions. Let f be continuous and bounded (|f | < M ) and let ǫ > 0 be a real number. Then, since the measure µ is finite, there exist real numbers r 1 , . . . , r k ∈ [-M, M ] (with r 1 = -M and r k = M ) such that |r i+1 -r i | < ǫ for all i = 1, . . . , k -1 and µ(f -1 ({r i })) = 0 for all i = 1, . . . , k. It follows that for i = 1, . . . , k -1 the sets C i = f -1 (]r i , r i+1 [) are all continuity open sets.

Hence the function f ǫ = k-1 i=1 r i 1 C i satisfies f -f ǫ ∞ ≤ ǫ and then the result follows by density.


Sofia Fenni
email: sofia.fenni@gmail.com

Fayrouz Haddad
email: fayrouz.haddad@im2np.fr

Antonio Jaomiary
email: jaomiaryantonio@yahoo.fr

Samar S Yazdani
email: ssamaryazdani@gmail.com

Frank Elliot Sahoa

Lucius Ramifidisoa
email: ramifidisoa@yahoo.fr

Mathieu Guerin
email: mathieu.guerin@im2np.fr

Wenceslas Rahajandraibe
email: wenceslas.rahajandraibe@im2np.fr

Blaise Ravelo
email: blaise.ravelo@yahoo.fr

Investigation on four-port mono-capacitor circuit with high-pass negative group delay behavior

Keywords: Mono-capacitor circuit, four-port topology, high-pass (HP) negative group delay (NGD), NGD function, NGD analysis, NGD specifications, S-matrix modelling

Most of the negative group delay (NGD) circuit recent investigation were focused on the classes of lowpass and bandpass categories. Few research works are currently available in the literature on the typical high-pass (HP) NGD function. This paper introduces an original HP-NGD circuit theory of four-port passive topology constituted by a single lumped capacitor. The S-matrix equivalent model of the innovative topology is established from the admittance matrix. Then, the basic frequency analytical responses are expressed. Then, the GD model is derived in function of the capacitor. It was found analytically that the four-port mono-capacitor passive circuit is susceptible to behave originally as an unfamiliar HP-NGD function. An innovative HP-NGD analysis is explored in addition to the expression of synthesis design equation in function of the expected NGD cut-off frequency which is linked to NGD optimal value.

The validity of the established HP-NGD circuit theory is verified with a proof of concept (POC). As expected, the HP-NGD behavior showing a good correlation between analytical model and simulation is obtained.

INTRODUCTION

In addition to the noise, the delay effect is one of the most penalizing effect which limits the electronic and communication system performances [START_REF] Groenewold | Noise and group delay in active filters[END_REF][START_REF] Myoung | Effect of group delay in RF BPF on impulse radio systems[END_REF][START_REF] Agrawal | Multi-band RF time delay element based on frequency translation[END_REF]. An unfamiliar negative group delay (NGD) equalization and neutralization solution against the delay effect was recently initiated [START_REF] Xiao | Negative Group Delay Circuits and Applications: Feedforward Amplifiers, Phased-Array Antennas, Constant Phase Shifters, Non-Foster Elements, Interconnection Equalization, and Power Dividers[END_REF][START_REF] Ravelo | Neutralization of LC-and RC-Effects with Left-Handed and NGD Circuits[END_REF]. The NGD solution is also promising for the distorted signal reconstruction [START_REF] Ravelo | Reconstruction Technique of Distorted Sensor Signals with Low-Pass NGD Function[END_REF]. The NGD function was more generally suggested to compensate for communication system delay [START_REF] Broomfield | Broadband Negative Group Delay Networks for Compensation of Oscillators, Filters and Communication Systems[END_REF], enhanced bandwidth feedback microwave amplifiers [START_REF] Choi | Efficiency enhancement of feedforward amplifiers by employing a negative group delay circuit[END_REF][START_REF] Choi | Bandwidth enhancement of an analog feedback amplifier by employing a negative group delay circuit[END_REF] and design of reconfigurable series feed network for squint-free antenna beamforming [START_REF] Zhu | Reconfigurable Series Feed Network for Squint-free Antenna Beamforming Using Distributed Amplifier-Based Negative Group Delay Circuit[END_REF]. However, because of particularly counterintuitive NGD effect, further research work is needed to open the NGD circuit design to non-specialist electronic engineers.

The existence of negative group delay (NGD) effect is one of most curious questions of physicists in the twentieth century. The unfamiliar NGD effect was observed in first time based on optical experimentation with negative group velocity (NGV) media [START_REF] Chu | Linear Pulse Propagation in an Absorbing Medium[END_REF][START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF]. Diverse unfamiliar NGD investigations with optical system were conducted by showing counterintuitive superluminal pulse propagation [START_REF] Munday | Observation of Negative Group Delays within a Coaxial Photonic Crystal Using an Impulse Response Method[END_REF][START_REF] Ségard | Two-pulse interference and superluminality[END_REF]. In addition, for further physical understanding, theoretical study of NGD effect was also developed via transfer function elaboration [START_REF] Munday | Observation of Negative Group Delays within a Coaxial Photonic Crystal Using an Impulse Response Method[END_REF][START_REF] Ségard | Two-pulse interference and superluminality[END_REF]. The curious question about the NGD function existence in the electronic circuits attracted the attention of some design researchers [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF][START_REF] Nakanishi | Demonstration of Negative Group-delays in a Simple Electronic Circuit[END_REF][START_REF] Kitano | Negative Group-delay and Superluminal Propagation: An Electronic Circuit Approach[END_REF][START_REF] Munday | Superluminal Time Advance of a Complex Audio Signal[END_REF][START_REF] Kandic | Asymptotic Limits of Negative Group Delay in Active Resonator-Based Distributed Circuits[END_REF].

Therefore, theoretical and experimental studies of low frequency (LF) NGD circuits were performed by using R, L, C and operational amplifier [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF]. The extraordinary experimentation of smoothed pulse signal in millisecond timeadvance and in superluminal propagation was performed [START_REF] Nakanishi | Demonstration of Negative Group-delays in a Simple Electronic Circuit[END_REF][START_REF] Kitano | Negative Group-delay and Superluminal Propagation: An Electronic Circuit Approach[END_REF][START_REF] Munday | Superluminal Time Advance of a Complex Audio Signal[END_REF]. It was emphasized that the NGD effect does not contradict the causality principle [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF].

Despite the notable limitation reported on the NGD properties [START_REF] Kandic | Asymptotic Limits of Negative Group Delay in Active Resonator-Based Distributed Circuits[END_REF], the NGD effect existence in microwave frequencies was an open question [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Markley | Quad-Band Negative-Refractive-Index Transmission-Line Unit Cell with Reduced Group Delay[END_REF][START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF][START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Chaudhary | Tunable center frequency negative group delay filter using coupling matrix approach[END_REF][START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF][START_REF] Wu | Maximally Flat Negative Group Delay Circuit: A Microwave Transversal Filter Approach[END_REF][START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF][START_REF] Ravelo | Synthesis of RF Circuits with Negative Time Delay by Using LNA[END_REF][START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF][START_REF] Ravelo | Bandpass NGD TAN of Symmetric H-Tree with Resistorless Lumped-Network[END_REF]. Remarkable NGD microwave passive circuits were designed in early 2000s from metamaterial based negative refractive index and split ring resonator structures [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Markley | Quad-Band Negative-Refractive-Index Transmission-Line Unit Cell with Reduced Group Delay[END_REF][START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF]. However, the metamaterial based NGD passive circuits are significantly lossy and limit the opportunity of their applications. Emphatically, intensive research works were made on low attenuation loss NGD passive circuits last two decades [START_REF] Liu | Compact transmission-type negative group delay circuit with low attenuation[END_REF][START_REF] Chaudhary | Tunable center frequency negative group delay filter using coupling matrix approach[END_REF][START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF][START_REF] Wu | Maximally Flat Negative Group Delay Circuit: A Microwave Transversal Filter Approach[END_REF][START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF][START_REF] Ravelo | Synthesis of RF Circuits with Negative Time Delay by Using LNA[END_REF].

Nevertheless, the diversity of NGD circuit structures leads to wonder about the general topologies of the unfamiliar electronic function. Such a question motivates to initiate a fundamental circuit theory of NGD function inspired from filter function [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF].

The innovative concept of low-pass (LP) [START_REF] Ravelo | Reconstruction Technique of Distorted Sensor Signals with Low-Pass NGD Function[END_REF][START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF] and bandpass (BP) [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF][START_REF] Ravelo | Bandpass NGD TAN of Symmetric H-Tree with Resistorless Lumped-Network[END_REF][START_REF] Wan | Bandpass Negative Group Delay Theory of Fully Capacitive Δ-Network[END_REF] NGD circuit classification was established. Furthermore, we are constantly looking for the simplest NGD topologies by using resistive-capacitive network [START_REF] Ravelo | Reconstruction Technique of Distorted Sensor Signals with Low-Pass NGD Function[END_REF], resistorless lumped network [START_REF] Ravelo | Bandpass NGD TAN of Symmetric H-Tree with Resistorless Lumped-Network[END_REF] and capacitive-network [START_REF] Wan | Bandpass Negative Group Delay Theory of Fully Capacitive Δ-Network[END_REF]. In difference to the all existing NGD research work, we are developing a circuit theory of mono-capacitor and extremely simple topology in the present paper.

S-PARAMETER MODELLING OF THE MONO-CAPACITOR NGD TOPOLOGY

The present section describes the topological identification of the mono-capacitor NGD circuit. For the better understanding about the unfamiliar NGD aspect, the basic function will be defined. Then, the S-parameter model will be exploited to elaborate the NGD class identification, and analysis.

Topological description

We are expecting a first order NGD topology. Therefore, must be composed of at least a reactive component [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF].

Accordingly, we can choose the impedance Z a first order element. Before the NGD topological identification, we can consider the four-port H-shape circuit shown by Fig. 1. By denoting the Laplace variable, s, the proposed circuit can be assumed to be composed of central element represented by an impedance, Z(s)=1/(Cs) and four lumped identical resistances, R. To analyze the circuit, each port n={1,2,3,4} is connected to voltage source Vn with the input current In.

By using the Kirchhoff voltage and current laws (KVL and KCL), we can write the following relathinoships: These KVL and KCL will serve in the following paragraph to establish the admittance matrix model of our four-port circuit.

(1) (2) (3) (4)

Admittance matrix modelling

The matrix equivalent of these four linear relations is expressed as: [START_REF] Ravelo | Neutralization of LC-and RC-Effects with Left-Handed and NGD Circuits[END_REF] with:

(6)

1 4 1 4 ( ) ( ) ( ) ( ) V s V s RI s RI s - = - 2 3 2 3 ( ) ( ) ( ) ( ) V s V s RI s RI s - = - [ ] 1 2 1 1 4 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) V s V s RI s Z s I s I s RI s - = + + - 1 2 3 4 ( ) ( ) ( ) ( ) 0 I s I s I s I s + + + = [ ] [ ] 1 1 2 2 3 3 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) V s I s V s I s s V s I s V s I s é ù é ù ê ú ê ú ê ú ê ú G ´= W ế ú ê ú ê ú ê ú ë û ë û [ ] 1 0 0 1 0 1 1 0 1 1 0 0 0 0 0 0 - é ù ê ú - ê ú G = ê ú - ê ú ë û . ( 7
)
According to the system theory, the admittance matrix is linked to the input voltage and access current by the matrix relationship:

.

By algebraic identification between equation ( 5) and equation ( 8), we established that the admittance matrix of our four-port network:

.

Consequently, we have: [START_REF] Zhu | Reconfigurable Series Feed Network for Squint-free Antenna Beamforming Using Distributed Amplifier-Based Negative Group Delay Circuit[END_REF] This admittance matrix will be exploited in the next subsection to determine the S-matrix of the NGD circuit.

S-matrix modelling

Let us consider the four-port black box introduced in Fig. 2. The S-matrix equivalent model can be generally denoted by: [START_REF] Chu | Linear Pulse Propagation in an Absorbing Medium[END_REF] This S-matrix is referenced by the access port impedance R0=50 Ω. [ ]

0 0 0 0 ( ) ( ) 0 ( ) 1 1 1 1 R R R R s R Z s R Z s é ù ê ú - ê ú W = ê ú + - ê ú ë û [ ] 1 1 2 2 3 3 4 4 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) I s V s I s V s Y s I s V s I s V s é ù é ù ê ú ê ú ê ú ê ú = ế ú ê ú ê ú ê ú ë û ë û [ ] [ ] [ ] ( ) ( ) Y s s = W ´G [ ] [ ] 3 2 ( ) 2 ( ) 3 2 ( ) 2 ( ) 2 ( ) 3 2 ( ) 2 ( ) 3 2 ( ) ( ) 4 ( ) R Z s R R R Z s R R Z s R Z s R R R Z s R Z s R R Z s R R R Z s Y s R R Z s + - - -- é ù ê ú - + -- - ê ú ê ú - -- + - ê ú -- - - + ë û = + [ ]
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

S s S s S s S s S s S s S s S s S s S s S s S s S s S s S s S s S s

é ù ê ú ê ú = ê ú ê ú ë û
According to the S-matrix theory, we have the Y-to-S matrix transform: [START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF] with the 4-D identity matrix:

Substituting the admittance matrix given in equation ( 10) into relationship [START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF], we have the following symmetric matrix:

.

From this S-matrix, the NGD topological analysis can be elaborated.

NGD topological identification

By the reason of simplicity, let us take R=0 or replaced by short-circuited branches. Therefore, we have the reduced four-port topology depicted by Fig. 3 which is constituted only by a capacitive component C. The associated S-matrix

given by equation ( 14) is composed of elements of written as: . ( 16)

[ ] [ ] [ ] { } [ ] [ ] { } 1 4 0 4 0 ( ) ( ) ( ) S s Id R Y s Id R Y s - = - ´+ [ ] 4 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 Id é ù ê ú ê ú = ê ú ê ú ë û [ ] 1 2 2 3 2 1 3 2 2 3 1 2 3 2 2 1 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) N s N s N s N s N s N s N s N s N s N s N s N s N s N s N s N s S s D s é ù ê ú ê ú ê ú ê ú ë û = [ ] [ ] [ ] 2 1 0 0 2 0 0 3 0 0 0 0 ( ) 2 2 ( ) ( ) ( ) ( ) 2 ( )
( ) 2( ) ( ) N s R R R Z s R N s R R R N s R R R Z s D s R R R R Z s ì = + + - ï = + ï í = + + ï ï = + + + î [ ] 1 0 2 0 3 0 0 ( ) ( ) ( ) 2 ( ) ( ) 2 ( ) N s R N s R N s R Z s D s R Z s ì = - ï = ï í = + ï ï = + î
By choosing the impedance as a capacitor component, , the previous coefficients can be simplified as transfer functions:

. [START_REF] Kandic | Asymptotic Limits of Negative Group Delay in Active Resonator-Based Distributed Circuits[END_REF] The NGD analysis of this transfer function will be explored in the following section.

NGD ANALYSIS OF THE IDENTIFIED PASSIVE TOPOLOGY

The NGD analysis of the mono-capacitor based circuit topology will be explored theoretically in the present section.

The expressions of the previously introduced NGD characteristics will be expressed in function of the capacitor component.

Basic frequency responses needed for the NGD analysis

The NGD study of a circuit depends essentially on the analysis of transmission coefficient. By taking the Laplace variable versus angular frequency, , the transmission coefficient (with integer subscript, m≠n) can be expressed as:

.

The basic elements to explore this response are transmittance, T(w), and phase, j(w) which are given by respectively:

[ ]

0 1 0 ( ) ( ) ( ) 2 ( ) nn R N s S s D s R Z s - = = + [ ] 3 0 41,32 0 ( ) 2 ( ) ( ) ( ) ( ) 2 ( ) N s R Z s T s S s D s R Z s + = = = + ( ) 1/ ( ) Z s Cs = 0 0 ( ) 2(1 ) nn R Cs S s R Cs - = + 0 0 2 ( ) 2 (1 )
L R Cs T s R Cs

+ = + s j = w ( ) ( ) mn T j S j = w w (22)
.

We can derive from this last expression, according to circuit theory, the GD defined by the following equation:

. ( 24
)
Knowing these fundamental elements, we can explore the unfamiliar NGD response. The basic characteristics of this response can be introduced as described in the following paragraph.

HP-NGD ideal specifications

The unfamiliar HP-NGD analysis of our four-port mono-capacitor passive topology must be performed with respect to specific parameters linked to the frequency responses. In this case of study, in difference to most of classical electronic circuit, the main parameters of the characterization are essentially the GD and the magnitude. Subsequently, Fig. 4(a)

and Fig. 4(b) introduce ideal plots associated the GD and transmission coefficient magnitude. We remind that if GD0<0, the circuit can be classified as LP-NGD topology. To operate as a HP NGD circuit, the LF GD value defined in equation ( 24) is equal to:

. ( 25
) ( ) ( ) T T j = w w [ ] ( ) arg ( ) T j = j w w ( ) ( ) GD ¶ = - ¶ j w w w ( 0) 0 p GD GD » = > w
Acting as passive circuit, at these LF, the magnitude is denoted by: .

The NGD cut-off frequency which is the root of equation: [START_REF] Shao | A Compact Dual-Band Negative Group Delay Microwave Circuit[END_REF] is denoted by: .

The positive GD bandwidth is defined in the LF, . Particularly to HP-NGD function, in the NGD frequency band, , we have:

(29)
As illustrated in Fig. 4(a), the GD response of our HP-NGD response is specified by an optimal frequency, ,

where we have the GD minimal value:

. ( 30
)
As seen in Fig. 4(b), in the NGD frequency band, , we can define the attenuation losses by: (

.

As concrete illustration of this unfamiliar and ideal HP-NGD function, let us consider in the following section deeper NGD analysis of circuit introduced in Fig. 1.

Frequency responses of the considered passive topology

This transmission coefficient can be identified by the canonical form: [START_REF] Wan | Bandpass Negative Group Delay Theory of Fully Capacitive Δ-Network[END_REF] by taking the parameters:

.

Substantially, the transmittance associated to our 4-port circuit defined in equation ( 22) is given by: .

By definition of equation ( 23), the corresponding phase response is equal to:

(36)

0 ( 0) 1 T T » = £ w ( ) 0 GD = w 2 n n f = w p n < w w n > w w ( ) 0 GD < w 2 o o f = w p [ ] min ( ) 0 o o GD GD = = < w w n > w w ( ) n n T T = = w w ( ) o o T T = = w w 1 2 ( ) 1 a s T s a s + = + 0 a R C = 2 2 2 2 1 4 ( ) 1 a T a + = + w w w 1 2 ( ) ( ) ( ) = + j w j w j w with (37) . ( 38
)
According to definition [START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF], the GD can be formulated as:

. ( 39
)
It can be understood from this expression that the circuit behaves as a HP-NGD topology.

HP-NGD characterization

The LF GD value defined in equation ( 25) is equal to:

. ( 40
)
The NGD cut-off frequency associated to equation ( 27) by means of expression (39) is given by: .

At the cut-off frequency, we have attenuation:

.

Moreover, the optimal frequency is equal to:

. ( 43
)
It means that the ratio between the cut-off and optimal frequencies is equal to:

. ( 44
)
Where the mono-capacitor circuit reaches the GD minimum which is obviously a negative quantity:

. ( 45
)
At this optimal frequency, the attenuation is equal to:

. ( 46
)
The HP-NGD characteristic equation can be formulated by the NGD value and NGD cut-off frequency relationship: 5 highlights the logarithmic variation plot of the optimal GD versus the two characteristic frequencies from -1 ns to -1 ms. We can see that the two frequencies vary from 0.6436 Hz to 6.436 MHz, and 2.2742 Hz to 2.2742 MHz, respectively. By considering a POC, the next section will be focused on the validation study of the elaborated HP-NGD circuit theory.

. ( 47
) 1 ( ) arctan 2 a ae ö = ç ÷ è ø w j w 2 ( ) arctan( ) a = j w w 2 2 2 2 2 2 (2 ) ( ) (1 )(4 ) a a GD a a - = + + w w w w 0 2 2 p R C a GD = = 0 1 1 2 2 2 2 n f a RC = = p p 1/ 2 n T = 3 2 2 2 o f a + = p 2(3 2) o n f f = + 2 2 1 3 o GD a ae ö = - ç ÷ è ø 2 2 o T = 2 2 3 6 2 (2 2 3)( 3 2 2) 6 n o o o f GD f GD ì - = ï ï í ï - + = ï î p p Fig.

HP-NGD BEHAVIOR VALIDATIONS

The present section is focused on the mono-capacitor HP-NGD circuit validation. Doing this, the response calculated from MATLAB® will be compared with commercial tool-based S-parameter simulation.

Parametric investigation with respect to the capacitor

A general view on the GD responses of our circuit was investigated by considering the capacitor C with minimal and maximal values, Cmin=1 pF and Cmax=1 µF. The parametric study was performed in the frequency range swept between 1 kHz and 1 GHz. The magnitude and GD responses were computed from analytical models expressed by equation (35) and equation (38), respectively. Therefore, we obtain the computed results represented by the mappings of respectively. We can remark that:

-The reflection coefficient magnitude, S11, increases with the capacitor and frequency, -And the transmission coefficient one, S21, varies inversely. As shown by Fig. 7(a), the positive value of GD is located in the zone of higher value of capacitor and lower value of the frequency. Then, the NGD zone is observed when the frequency is much higher. We can understand from the three maps that the circuit is susceptible to behave as HP-NGD function with the navy-blue surface area of the mapping. Nevertheless, it can be understood that the optimal NGD frequency appears around a characteristic medium value of the capacitor. The absolute value of the NGD optimum is decreasing with the frequency. For the better validation of the HP-NGD aspect, a comparative study between calculation and simulation was performed. The next subsection reports the POC circuit design.

Design description of the POC circuit

Our validation approach is based on the comparison between the MATLAB numerical computation of analytical expressions given in equations ( 35), ( 36) and (39) with the simulation results from the commercial tool of electronic circuit simulator "Advanced Design System (ADS®)" from Keysight Technologies®. The schematic design of the mono-capacitor circuit designed in the ADS® environment is illustrated by Fig. 8. It acts as a four-port circuit with terminals operating under reference impedance R0=50 W. A lumped capacitor with nominal value C=22 pF was considered as an arbitrary test component. 3. As reported in this recapitulative table, the calculation and simulation differences are almost zero (indeed zero for frequency). This fully agrees with the behavior of resonant circuits, for which the phase corresponding to the NGD center frequency is equal to zero. The NGD phenomenon appears explicitly at the center frequency with NGD value of about GDn=-37 µs over the NGD bandwidth, BWn=10 kHz. In other words, this bandwidth corresponds to about 8% of the NGD center frequency. The RLCP circuit attenuation is of about 9.54 dB at the NGD center frequency. We can point out that this result satisfies the condition for maximizing the FoM as earlier predicted in Fig. 2. After this spectacular validation results, we are wondering about a curious question ever being answered in the previous NGD investigation available in the literature Simu. Model about the influence of the R, L and C component tolerances on the NGD performances. The answer to this question constitutes the main object of the following section.

CONCLUSION

An original circuit theory on the HP-NGD four-port typical H-shape topology ever being done before is developed.

The considered topology is essentially comprised of a single mono-capacitor. In the future, we expect expanded applications of HP-NGD circuits with higher number of port topology. In this optics, the present NGD investigation approach could contribute to the processing of transmission signals requiring optimization in terms of transmission delay.

Figure 1 :

 1 Figure 1: R and Z impedance based four-port circuit.

Figure 2 :

 2 Figure 2: Four-port S-parameter model.

Figure 3 :

 3 Figure 3: Mono-capacitor four-port circuit.

Figure 4 :

 4 Figure 4: NGD specifications: (a) GD and (b) transmission coefficient magnitude.

Figure 5 :

 5 Figure 5: Variation of GDo versus (a) fn and (b) fo.

  coefficients versus couple (frequency, capacitor) displayed by Fig. 6(a) and Fig. 6(b),

Figure 6 :

 6 Figure 6: Mappings of (a) S11 and (b) S21 versus couple (frequency, capacitor).

Figure 7 :

 7 Figure 7: Mapping of GD versus couple (frequency, capacitor) in the frequency bands, (a) [1 kHz, 200 kHz], (b) [200 kHz, 20 MHz] and (c) [20 MHz, 1 GHz].

Figure 8 : 4 . 3 .

 843 Figure 8: ADS® design of the simulated NGD passive mono-capacitor 4-port circuit.

Figure 9 :Figure 10 :Figure 11 :

 91011 Figure 9: Comparison between the calculated and simulated (a) S11 and (b) S21 magnitudes of the POC HP-NGD circuit shown in Fig. 8.

  The S-parameter model of the mono-capacitor topology is established from the admittance matrix expression. The theoretical development of the HP-NGD analytical approach is elaborated. An innovative characterization of the HP-NGD specifications in function of the C component value is introduced. The main specifications of the HP-NGD function as the NGD cut-off frequency, NGD optimal frequency and optimal NGD value, and attenuation were investigated in function of frequency variations and the capacitor element. The analytical expressions of each NGD specification were established. To validate the HP-NGD theory, results of from analytical and simulation with the ADS® commercial tool were discussed. It was demonstrated from parametric analyses that the four-port topology is susceptible to behave as an HP-NGD function. Moreover, calculated and simulated results in very good correlation are obtained.

Table 2 :

 2 Targeted NGD specifications

	Approach	fn (kHz)	fc1 (kHz)	fc2 (kHz)	BW (kHz)	Tn (dB)	GDn (µs)	FoM
	Calc.	125	120.1	130.09	9.9901	-9.5424	-36.792	0.12252
	Simu.	125	120.1	130.1	10	-9.541	-36.78	0.1225
	Gap (%)	0	0	0.0077	0.0991	0.32	0.0326	0.0163

Table 3 :

 3 Comparison between the calculated and simulated NGD parameters


Fayu Wan
email: fayu.wan@nuist.edu.cn

Xiaoyu Huang

Konstantin Gorshkov
email: k.gorshkov@list.ru

Bogdana Tishchuk
email: b.tishchuk@list.ru

Xiaofeng Hu

Frank George Chan
email: george.chan@asmpt.com

Elliot Sahoa

Sahbi Baccar
email: sbaccar@cesi.fr

Mathieu Guerin
email: mathieu.guerin@im2np.fr

Wenceslas Rahajandraibe
email: wenceslas.rahajandraibe@im2np.fr

Frank Elliot Sahoa

Blaise Ravelo
email: blaise.ravelo@yahoo.fr

High-pass NGD characterization of resistive-inductive network based low-frequency circuit

Keywords: High-pass (HP) negative group delay (NGD), RL-network, circuit theory, NGD measurement technique

teaching and research institutions in France or abroad, or from public or private research centers.

1/ INTRODUCTION

Most of communication devices are regularly victim of undesirable delay effects [START_REF] Groenewold | Noise and Group Delay in Active Filters[END_REF][START_REF] Myoung | Effect of Group Delay in RF BPF on Impulse Radio Systems[END_REF]. There are different sources of delays at different level of communication systems. For example, at electronic component levels the interconnect complexity induces signal delays and distortions [START_REF] Kang | A global delay model for domino cmos circuits with application to transistor sizing[END_REF][START_REF] Ravelo | Delay Modelling of High-Speed Distributed Interconnect for the Signal Integrity Prediction[END_REF] Because of such imperfection, the leading and tailing edges of pulse communication signals can be significantly distorted [START_REF] Bukhman | On the distortion of the rising edge of a carrier-free signal[END_REF][START_REF] Bukhman | On the Distortion of the Leading Edge of a Quasi-Monochromatic Signal in a Resonantly Absorbing Medium[END_REF]. Against the delay, an innovative synthesis of delay-insensitive circuits was introduced [START_REF] Li | Optimised Synthesis of Delay-Insensitive Circuits Using Time-Sharing[END_REF]. For some cases of communication systems, the delay effects are exploited to identify the communication channel [START_REF] Narasimhan | Channel Blind Identification Based on Cyclostationarity and Group Delay[END_REF], to design recirculating delay line [START_REF] Heyde | Theoretical Methodology for Describing Active and Passive Recirculating Delay Line Systems[END_REF], to design a transmission line (TL) delay-based RFID tag [START_REF] Vemagiri | Transmission Line Delay-Based Radio Frequency Identification (RFID) Tag[END_REF] and also to improve the gain-bandwidth product of amplifiers [START_REF] Alves | A time-delay technique to improve GBW on negative feedback amplifiers[END_REF].

An orthodox technique of delay equalization, which is less familiar to most electronic engineers, based on the use of negative group delay (NGD) function was also introduced for the interconnect effect [START_REF] Ravelo | Theory and circuit modelling of baseband and modulated signal delay compensations with low-and band-pass NGD effects[END_REF], the filter phase linearization [START_REF] Shao | A group-delay-compensation admittance inverter for full-passband self-equalization of linear-phase band-pass filter[END_REF] and unconditional stable active filter [START_REF] Zhang | Unconditionally Stable Non-Foster Element Using Active Transversal-Filter-Based Negative Group Delay Circuit[END_REF]. However, compared to the other electronic functions (filter, amplifier, antenna, oscillator, power combiner/divider, …), the unfamiliar NGD ones are less familiar to most electronic design engineers. For this reason, the present study introduces a further investigation on typical high-pass (HP) NGD circuit by using resistive-inductive (RL) passive network. Before the exploration of the HP-NGD developed study, we propose to start with the state art.

The NGD function was initially experimented with negative group velocity (NGV) medium at optical wavelengths [START_REF] Ségard | Observation of Negative Velocity Pulse Propagation[END_REF][START_REF] Macke | Propagation of light-pulses at a negative group-velocity[END_REF][START_REF] Bukhman | On the Principle of Causality and Superluminal Signal Propagation Velocities[END_REF][START_REF] Bukhman | On the propagation velocity of a wave packet in an amplifying medium[END_REF][START_REF] Bukhman | On the reality of supraluminal group velocity and negative delay time for a wave packet in a dispersion medium[END_REF][START_REF] Dogariu | Transparent anomalous dispersion and superluminal lightpulse propagation at a negative group velocity[END_REF]. The NGD effect was investigated as abnormal and superluminal phenomena [START_REF] Bukhman | On the Principle of Causality and Superluminal Signal Propagation Velocities[END_REF][START_REF] Bukhman | On the reality of supraluminal group velocity and negative delay time for a wave packet in a dispersion medium[END_REF][START_REF] Dogariu | Transparent anomalous dispersion and superluminal lightpulse propagation at a negative group velocity[END_REF]. Then, the NGD research topic attracted some of electronic and RF and microwave engineering researchers [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Markley | Quad-Band Negative-Refractive-Index Transmission-Line Unit Cell with Reduced Group Delay[END_REF][START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF][START_REF] Bukhman | On the Negative Delay Time of a Narrow-Band Signal as It Passes through the Resonant Filter of Absorption[END_REF][START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF][START_REF] Nakanishi | Demonstration of Negative Group-delays in a Simple Electronic Circuit[END_REF][START_REF] Kitano | Negative Group-delay and Superluminal Propagation: An Electronic Circuit Approach[END_REF][START_REF] Munday | Superluminal Time Advance of a Complex Audio Signal[END_REF][START_REF] Mao | High fidelity of electric pulses in normal and anomalous cascaded electronic circuit systems[END_REF]. Design and test of microwave NGD circuits with metamaterial negative refractive index (NRI) structures were performed [START_REF] Eleftheriades | Transmission Line for Negative Refractive Index Media and Associated Implementations without Excess Resonators[END_REF][START_REF] Siddiqui | Periodically Loaded Transmission Line With Effective Negative Refractive Index and Negative Group Velocity[END_REF][START_REF] Markley | Quad-Band Negative-Refractive-Index Transmission-Line Unit Cell with Reduced Group Delay[END_REF][START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF]. Based on the analogy between the transfer function of 3-D and 2-D structures and 1-D microstrip circuits, topologies of RLC-resonant cells analog to the split ring resonator cell were identified [START_REF] Monti | Negative Group Velocity in a Split Ring Resonator-Coupled Microstrip Line[END_REF][START_REF] Bukhman | On the Negative Delay Time of a Narrow-Band Signal as It Passes through the Resonant Filter of Absorption[END_REF]. In parallel to the development of microwave circuit design, innovative topologies of low-frequency NGD circuit using R, L, C and operational amplifier components were proposed [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF][START_REF] Nakanishi | Demonstration of Negative Group-delays in a Simple Electronic Circuit[END_REF][START_REF] Kitano | Negative Group-delay and Superluminal Propagation: An Electronic Circuit Approach[END_REF][START_REF] Munday | Superluminal Time Advance of a Complex Audio Signal[END_REF]. Different simulations and experimental demonstrations of NGD effect [START_REF] Nakanishi | Demonstration of Negative Group-delays in a Simple Electronic Circuit[END_REF][START_REF] Kitano | Negative Group-delay and Superluminal Propagation: An Electronic Circuit Approach[END_REF][START_REF] Munday | Superluminal Time Advance of a Complex Audio Signal[END_REF][START_REF] Mao | High fidelity of electric pulses in normal and anomalous cascaded electronic circuit systems[END_REF] were successfully realized. Thus, despite the implicit counterintuitive effect, it was emphasized that the NGD function does not contradict the causality [START_REF] Mitchell | Negative group delay and ''fronts'' in a causal system: An experiment with very low frequency bandpass amplifiers[END_REF][START_REF] Mitchell | Causality and Negative Group-delays in a Simple Bandpass Amplifier[END_REF]. Last two decades, different design of active circuits based on bilateral gain-compensated [START_REF] Kandic | Bilateral Gain-Compensated Negative Group Delay Circuit[END_REF], current-feedback operational-amplifier and uses grounded resistors and capacitors [START_REF] Abuelma | A new CFOA-based negative group delay cascadable circuit[END_REF], and broadband switch-less bi-directional amplifier [START_REF] Meng | A Broadband Switch-Less Bi-Directional Amplifier with Negative-Group-Delay Matching Circuits[END_REF] circuits were proposed.

Despite the proposed various circuits, the meaning of the NGD function remains an open question. For the better understanding, the NGD theory based on the similitude between the filter behavior [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF][START_REF] Ravelo | On low-pass, high-pass, bandpass, and stop-band NGD RF passive circuits[END_REF] was initiated. The concept of low-pass (LP) [START_REF] Ravelo | First-order low-pass negative group delay passive topology[END_REF] and high-pass (HP) [START_REF] Ravelo | High-Pass Negative Group Delay RC-Network Impedance[END_REF] NGD functions was initiated. But so far, few research works are available in the literature on the HP-NGD topology. For this reason, the present paper is focused on the investigation on the HP-NGD topology using RL-network.

The paper is organized in three main sections:

• Section 2 defines analytically the main specifications of the HP-NGD ideal responses by considering illustrative representative GD and magnitude diagrams. Then, the voltage transfer function (VTF) of the RLnetwork based passive circuit is formulated.

• By examining the VTF, Section 3 develops the theorization of the RL-network based HP-NGD passive circuit.

The synthesis method allowing to determine the RL-circuit resistor and inductor parameters is elaborated in function of the desired HP-NGD specifications.

• Section 4 describes the prototypes of the fabricated HP-NGD passive circuits as a proof-of-concept (POC) are described. Then, the experimental setup of the HP-NGD prototype is introduced.

• Section 5 examines the experimental validation results. The comparisons of calculated, simulated and measured results will be discussed.

• Section 6 is the final conclusion.

2/ HIGH-PASS (HP) NGD THEORETICAL SPECIFICATIONS

The present section elaborates the theoretical approach of HP-NGD circuit. After the ideal specifications, the analytical formulations of the NGD existence condition in function of the constituting RL-network parameters will be developed.

2-1/ Basic analytical parameters

The proposed NGD circuit is based on the diagram shown by According to the circuit and system theory, the VTF is traditionally defined by:

() () () out in Vs Ns Vs = ( 1
)
with s=jω is the Laplace variable. By definition, the corresponding transmittance isochrone is written as:

    ( ) ( ) ( ) N j e N j j m N j =  +     (2)
with () ez  and () mz  are real and imaginary parts. By definition, the associated magnitude is given by:

    22 ( ) ( ) ( ) ( ) N N j e N j m N j = =  +      . (3)
The associated phase is expressed by:

      () ( ) arg ( ) arctan () m N j Nj e N j   ==         . ( 4
)
The GD derived analytically from this transmission phase is defined by:

() () GD - =     . ( 5
)
Knowing these basic parameters, the specifications of the HP-NGD response are defined in the following subsection.

2-2/ Definition of HP-NGD response

An electronic circuit can be assumed as a HP-NGD function if the GD at very low-frequencies (VLFs), defined by 0   , satisfies the condition:

( 0) 0 GD   . ( 6
)
The associated cut-off angular frequency, ωn, is the root of equation:

( ) 0 n GD =  . ( 7
)
This means that we should satisfy the inequality conditions:

( ) 0
( ) 0 n n GD GD        (8)
as explained by the ideal diagram of Fig. 2(a). In addition to the GD response, the VTF magnitude response as shown in Fig. 2(b) must be specified. For example, the LP-NGD circuit can be designed by expecting a desired attenuation:

0 ( 0) 1 NN  =   . ( 9
)
As materialization of this ideal specification, a topology of RL-network based circuit will be analyzed in the following subsection.

2-3/ HP-NGD analysis

The present HP-NGD analysis is theoretically inspired from the LP-NGD one [START_REF] Ravelo | Similitude between the NGD function and filter gain behaviours[END_REF][START_REF] Ravelo | On low-pass, high-pass, bandpass, and stop-band NGD RF passive circuits[END_REF][START_REF] Ravelo | First-order low-pass negative group delay passive topology[END_REF][START_REF] Ravelo | High-Pass Negative Group Delay RC-Network Impedance[END_REF] from the VTF exploration. The previous equation implies the transmittance isochrone as a complex number expressed as:

() () () m mm R R jL Nj R R jL R R + = ++    . ( 11
)
The magnitude corresponding to this VTF transmittance can be written by:

22 2 2 2 () ( ) ( ) ( ) ( ) ( ) m mm R R L N N j R R L R R + == ++    . ( 12
)
Based on equation ( 3), the associated phase is given by:

() ( ) arctan arctan m m L R R L R R R  +  =-       . ( 13
)
The GD is calculated from this equation by means of the mathematical definition of relation (4):

2 2 2 2 2 2 2 2 2 2 2 2 () () ( ) ( ) mm mm LR R R L R R GD R L R R L R R  -+  =  + + +     . ( 14
)
Knowing these fundamental expressions, we can make the NGD analysis by considering the particular frequencies as reported in the following section.

3/ HP-NGD ANALYSIS AND SYNTHESIS METHOD

The existence condition of HP-NGD behavior can be generally demonstrated by the calculation of the GD established previously at different particular frequencies. The following paragraphs examines the analytical investigation at very low frequencies (VLFs) and the NGD cut-off frequency.

3-1/ VLF analysis of the RL-network based topology

The most intuitive analysis of the RL-network topology understudy can be performed at VLF where ω≈0.

Accordingly, the VTF magnitude expressed in equation ( 12) is transformed as:

0 ( 0) m m R NN RR = = = +  . ( 15
)
Always at VLF (ω≈0), the GD given by equation ( 14) becomes:

0 ( 0) m L GD GD R =  =  . ( 16
)
We can see that this GD is always positive, 0 0 GD  . This analytical finding explains the necessary condition of the HP-NGD. The next step of the demonstration must be the existence of the NGD cut-off frequency.

3-2/ Analysis at NGD cut-off frequency

The NGD cut-off frequency is the solution of equation ( 7) by considering the GD formulated in equation ( 14).

Accordingly, we have the quantity versus the RL-network parameters:

m n m RR L R R = +  . ( 17
)
Substituting this cut-off frequency into the attenuation established in equation ( 12), we have:

() m n m R N RR = +  . ( 18
)
For the positive real coefficient, x>1, we can demonstrate that the GD at the multiple of angular frequency,

n x , is equal to: 2 2 2 2 (1 ) ( ) () (1 ) (1 )
m n mm x L R R GD x R x R x R x R -+ =     + + + +      . ( 19
)
Under the hypothesis x>1, it can be understood from this expression that this GD is always negative:

( ) 0 n GD x   . ( 20
)
Moreover, the attenuation of the VTF magnitude written in equation ( 12) at this angular frequency, n x , is formulated by:

2 22 (1 ) () ( ) (1 )
mm n mm R R x R Nx R R x R x R  ++  =  + + +   . ( 21
)
Knowing the positive GD established by equation ( 16) and the existence of the cut-off frequency given by equation [START_REF] Bukhman | On the Principle of Causality and Superluminal Signal Propagation Velocities[END_REF], this analytical result illustrates the HP-NGD behavior of the RL-network topology introduced by Fig. 3.

3-3/ NGD optimal frequency

In opposite to the case of LP-NGD function investigated in [START_REF] Ravelo | First-order low-pass negative group delay passive topology[END_REF], the HP-NGD circuit should present an optimal angular frequency,

x  , where:

  ( ) min ( ) x GD GD =  . ( 22
)
This NGD optimal frequency should verify the relation:

() 0 x GD =  =     . ( 23
)
After resolution of this equation, we can demonstrate that the following optimal frequency written as:

3 ( 2 )( ) ( ) () m m m m m x m R R R R R R R R R L R R  + + + +  = +  . ( 24
)
For the sake of mathematical simplification, we can assume that this optimal angular frequency in function of NGD cut-off frequency, for example, by taking:

xn x =  . (25)
In other words, by means of equation ( 17) and equation ( 24) of characteristic angular frequencies, the real coefficient can be expressed in function of the HP-NGD circuit resistive parameters as:

3 ( 2 )( ) ( ) m m m m x n m R R R R R R R x RR  + + + +  == +   . ( 26
)
The optimal GD written in relation ( 4) can be determined by choosing:

( ) ( ) 0 xn GD GD x =  . ( 27
)
Under the same approach, the attenuation of the RL-network topology at the optimal angular frequency can be formulated as:

( ) ( ) 1 xn N N x =  . ( 28
)

3-4/ Synthesis method of HP-NGD circuits

The proposed synthesis method of HP-NGD circuit consists in calculating the components R, Rm and L. in function of the desired specifications. Doing this, we need three independent equations to realize the HP-NGD circuit synthesis.

3-4-1/ Synthesis equations of the R and L parameters in function of the HP-NGD specifications

The inputs of the synthesis method calculation are the values of NGD specifications earlier indicated in Figs. 2 as:

• The NGD cut-off frequency:

2 nn f =  (29)
• The attenuation at the NGD cut-off frequency:

( ) 1 nn NN =  (30)
• And the NGD:

( ) 0 xx GD GD =  (31)
at the NGD optimal frequency.

The resistor, Rm, synthesis equation is obtained by inverting equation ( 18) which implies:

2 2 1 n m n NR R N = - . (32)
In addition, the inductor synthesis equation obtained from the cut-off frequency formulated in equation ( 17) is given by to the following equation:

2 m nm RR L f R R = +  . ( 33
)
Substituting this expression of resistor, Rm, into the cut-off frequency formulated in equation ( 17), the inductor equation becomes:

2 n n NR L f =  . ( 34
)
Therefore, substituting equation [START_REF] Kandic | Bilateral Gain-Compensated Negative Group Delay Circuit[END_REF] and equation [START_REF] Meng | A Broadband Switch-Less Bi-Directional Amplifier with Negative-Group-Delay Matching Circuits[END_REF] into equation [START_REF] Bukhman | On the reality of supraluminal group velocity and negative delay time for a wave packet in a dispersion medium[END_REF], it yields the following optimal GD equation:

2 2 2 2 2 2 2 2 (1 ) 2 1 (1 ) (1 ) (1 )
n x n n n n n x N R GD f N N x x N N x - =     - + + - + +      . ( 35
)
By inverting this last equation, the synthesis equation of resistor can be expressed as:

2 2 2 2 2 2 2 2 2 1 (1 ) (1 ) (1 )
(1 )

n x n n n n n f GD N N x x N N x R x N R     - + + - + +     = -  . ( 36
)
These synthesis formulae enable to elaborate some analytical characteristics of the circuit topology understudy.

3-4-2/ Characterization of HP-NGD circuit attenuation at the optimal frequency

The characterization aims to study the relation between the different attenuations and the ratio of the NGD optimal and cut-off frequency. The characteristic of the HP-NGD can be established in function of the chosen cut-off frequency, fn, attenuation, Nn, and the optimal GD, GDx. It is worth to note that the optimal attenuation, Nx, given by equation ( 21) can be expressed in function of the cut-off frequency attenuation, Nn, by the relationship:

2 22 (1 )
(1 )

nm x m N R x R N x R x R ++ = ++ . ( 37
)
The substitution of the resistor equation [START_REF] Kandic | Bilateral Gain-Compensated Negative Group Delay Circuit[END_REF] and equation [START_REF] Ravelo | On low-pass, high-pass, bandpass, and stop-band NGD RF passive circuits[END_REF] into the previous expression implies systematically this modified optimal attenuation:

2 2 2 2 2 2 2 1 (1 ) (1 )
(1 )

n n n x nn N N N x N x N N x - + + = - + + . ( 38
)
The numerical calculations of this optimal attenuation in function of Nn varied from -10 dB to -1 dB and the ratio

x varied from 1.1 to 10 gives the mapping displayed by Fig. 4(a). This 2-D cartographical mapping is practically useful to predict the optimal attenuation in function of the chosen parameters N n and x during the HP-NGD circuit synthesis. It can be understood from this cartography that Nx increases with Nn. However, as illustrated by the three curves for Nn={ -1 dB, -2 dB, -3 dB} shown in Fig. 4(b), the optimal attenuation is inversely proportional with the ratio x. To validate the developed HP-NGD circuit theory, an experimental demonstrator will be discussed in the next section.

4/ EXPERIMENTAL TEST DESCRIPTION

The present section describes the experimental validation of the HP-NGD theory of RL-network. After the POC description, the validation results will be discussed. 5(b). Table 1 summarizes the main specifications of the equipment's employed of the experimented HP-NGD circuits. The following subsection discusses and examines the results from the simulated, fabricated and tested HP-NGD circuit prototypes. After the consideration of the HP-NGD specifications, the ideal values of the simulated circuit are calculated.

4-1/ Experimentation technique

4-2/ Fabricated HP-NGD circuit prototypes

Table 2 addresses the references of the components constituting the two HP-NGD circuit prototypes. Then, the nominal values of the used components which have 5% relative tolerances are indicated in the right side of Table 2.

Table 2: Component values and references

With the fabricated HP-NGD circuit prototype, the formulation method of the NGD responses are indicated in the following subsection.

4-3/ Formulation of measured HP-NGD responses

The HP-NGD responses from the experimental setup introduced in the previous subsection are obtained from the input and output sine responses. The t-time dependent input sine signal can be expressed in function of the test frequency, fk and amplitude, Amax, by the equation: max ( ) sin( 2)

in k v t A f t =  . ( 39
)
Acting as a linear time-invariant, the associated output signal can be expressed by: max ( ) sin( 2)

out k v t B f t = +   . ( 40
)
The HP-NGD circuit measurement consists originally in sweeping the frequency fk of the input sine signal from fmin=1 kHz to fmax=50 kHz. During the test, we choose the frequency step equal to Δf=50 Hz. The input signal was windowed over the time width of Δt=40 µs. We recorded during the tests the amplitudes and phases of the input and output sine signals. Then, the measured amplitudes and phase shift of the measured circuit VTF were calculated from the classical equations:

max max () () () k measured k k Bf Tf Af = (41) ( ) ( ) measured k k ff  = . ( 42
)
Accordingly, the GD response is determined by:

1 1 ( ) ( ) () 2 (
)

measured k measured k measured k kk ff GD f ff - - - = -   (43)
where the integer k varies from 2 to k max . A max and B max are the input and output signal amplitudes:

max min max ff k Ent f -  =    . ( 44
)
In this equation, the quantity, Ent[x] presents the superior integer part of real x. By using the developed HP-NGD circuit measurement techniques, we obtained the results explored in the following section.

5/ HP-NGD FUNCTION VALIDATION RESULTS

This section introduces the discussion on the simulated and measured validation results of the POCs HP-NGD circuits introduced previously. The results of the two HP-NGD circuit prototypes will be investigated in the following subsections. VLFs and become negative above the cut-off frequencies. Table 3 indicates the comparisons of the HP-NGD circuit specifications. We can see that the NGD cut-off frequency is of about 4 MHz according to the calculation and simulation against 5 MHz according to the measurement result. The slight difference of the measured magnitude about 0.5 dB is essentially due to:

• The lumped component tolerances of the RL-network circuit prototype1,

• The imperfections of the PCB fabrications,

• And the systematic errors from the measurement techniques induced by the oscilloscopes and the used cable-connect networks. Such results correspond clearly to the behavior of HP-NGD function. Table 4 gives the comparative values of the calculated, simulated and measured specifications around the NGD cut-off and optimal frequencies. The differences between the experimental results and the other ones are caused by the same reasons as the previous case of study.

5/ CONCLUSION

A particular innovative study of unfamiliar lumped passive circuit is investigated. The study is using an RLnetwork based HP-NGD topology.

The basic specifications of the HP-NGD function are defined. The circuit analytical investigation is elaborated from the VTF expression. The detailed expressions of the NGD cut-off frequency and the optimal NGD value are

Fig. 1 .

 1 It is constituted by two-port with input voltage Vin and output voltage Vout. The NGD analysis is based on the GD response from the voltage transfer function (VTF).

Fig. 1 :

 1 Fig. 1: NGD circuit black box diagram.

Fig. 2 :

 2 Fig. 2: Ideal responses of VTF (a) GD and (b) magnitude.

Fig. 3 Fig. 3 :

 33 Fig. 3 introduces the RL-network based passive circuit under investigation. This passive topology is composed by a series Rm resistor shunted by series RL-network. It can be demonstrated from voltage divider principle that the VTF model of the considered RL-circuit is given by:

Fig. 4 :

 4 Fig. 4: (a) 2-D mapping and (b) 1-D plot of the optimal attenuation versus the NGD optimal-and cut-off frequency ratio.

Fig. 5 (

 5 Fig. 5(a) highlights the experimental setup diagram of the BP NGD POC. The LF signal generator, referenced AKIP 4126/4A, generates a periodic sine signal at the input of the circuit. The input and the output signals are observed on oscilloscope referenced AKIP 4122/3. The RL-network circuit is mounted on fabricated as printed circuit board. The LF signal generator and the oscilloscope are connected to the circuit under test (CUT) via BNC cables as illustrated in Fig. 5(b).Table 1 summarizes the main specifications of the equipment's employed of the

Fig. 5 :

 5 Fig. 5: (a) Illustrative diagram and (b) photograph of the HP NGD circuit experimental setup.

Fig. 6 (

 6 Fig. 6(a) represents the schematic of simulated HP-NGD circuit from the ADS® environment. This simulation panel contains the circuit design, the component values of the two POCs and also the type of considered AC simulation. The photographs of the tested HP-NGD circuit prototypes are displayed by Fig. 6(b) and Fig. 6(c).

Fig. 6 :

 6 Fig. 6: (a) Schematic and (b)-(c) photographs of prototypes of tested HP NGD circuits.

5- 1 /

 1 Fig. 7(a) and Fig. 7(b) represent the VTF magnitudes and phases of the calculated model ("Model", black solid line), simulated results ("Sim.", blue dotted line) and measured results ("Meas.", red dashed line) of tested circuit prototype1. The associated wide band and zoomed in narrow band GDs are displayed by Figs. 8. It can be pointed out that the results of calculation, simulation and measurement are particularly in good correlation. We can underline that the GD response confirm the HP-NGD function of the tested RL-circuit. The GDs are positive at

Fig. 7 :Fig. 8 :

 78 Fig. 7: Calculated, simulated and measured (a) magnitude and (b) phase responses of tested HP-NGD prototype1 with R1=330 Ω, Rm1=910 Ω and L1=10 mH.

Table 3 : HP-NGD specifications of prototype1 4 - 2 /

 342 Discussion on the results of HP-NGD circuit prototype2The present paragraph examines the validation results of the tested circuit prototype 2 . The results are similar to the previous case of prototype1. The HP-NGD function is validated by the behavior of the calculated, simulated and measured GD responses. Fig.9(a) and Fig.9(b) show the VTF magnitudes and phases of the calculated model, simulated results and measured results which are indicated by legends "Model" (black solid line), simulated "Sim." (blue dotted line) and "Meas." (red dashed line). Figs. 10 depict the associated GD responses. When we follow the curves following the increase of frequency, we can see that the plots of the modeled, simulated and measured GD start from positive value and becomes negative above the NGD cut-off frequency.

Fig. 9 :

 9 Fig. 9: Calculated, simulated and measured (a) magnitude and (b) phase responses of tested HP-NGD prototype2 with R2=910 Ω, Rm2=910 Ω and L2=10 mH.

Fig. 10 :

 10 Fig. 10: Calculated, simulated and measured (a) wide-band and (b) narrow-band GD responses of tested HP-NGD prototype2 with R2=910 Ω, Rm2=910 Ω and L2=10 mH.

Table 1 : Specifications of the test equipment's

 1

	Description	References	Parameter	Value
	Arbitrary function generator option	AKIP 4126/4A-X	Sampling rate Bandwidth	125 MS/s 25 MHz
	Digital oscilloscope	AKIP 4122/3	Sampling rate Bandwidth	2 GS/s 100 MHz

Table 4 : HP-NGD specifications of prototype2

 4

	Parameters	Name	Model	Simulation	Measurement
	Cut-off frequency	fn	9.78 MHz	9.801 MHz	11.7 MHz
	Attenuation	T(fn)	-2.9 dB	-2.94 dB	-2.99 dB
	Optimal frequency	fx	17 MHz	17.02 MHz	21 MHz
	Optimal GD	GDx	-1.25 µs	-1.25 µs	-0.92 µs
	Optimal attenuation	T(fx)	-4.37 dB	-4.37 dB	-4.81 dB

established. In addition, a particular original formulation of the HP-NGD topology attenuation at the cut-off and optimal frequencies are theoretically expressed in function of the R and L parameters. The synthesis equations enabling to determine these parameters in function of the desired HP-NGD specifications are expressed. In addition, the characterization of the attenuation is proposed.

The HP-NGD theory is validated by the feasibility study based on the design and fabrication of the POC prototype of PCBs. As expected, the calculated, simulated and measured magnitudes and GDs are good agreement. More importantly, the three validation results confirm the HP-NGD behavior of the frequency responses of the tested PCBs.