Nadjah Touati
email: nadjah.touati@ifsttar.fr

Charles Tatkeu
email: charles.tatkeu@ifsttar.fr

Thierry Chonavel
email: thierry.chonavel@telecom-bretagne.eu

Atika Rivenq

Design and Performances Evaluation of New Costas-Based Radar Waveforms with Pulse Coding Diversity

Costas codes are a variant of pulse compression waveforms, largely studied for their attractive time-frequency properties. Their «thumbtack-like » ambiguity function makes them highly suitable for delay and Doppler estimation, in radar and sonar applications.

However, this behavior depends heavily on the length of the code: the improvement in delay-Doppler resolutions and ambiguity function sidelobes level needs an increase in the size of the code. In this paper, designs that allow good performance without increasing the size of the code are proposed. They are based on a modification of Costas Codes by widening frequency separation between hops and replacing rectangular pulses by other waveforms. This will lead to a removal of autocorrelation function ACF grating lobes that normally appear when frequency separation is increased. The originality of the work lies in the proposal of diversified pulse waveforms, such as phase codes, Slepian sequences and other Costas codes, to encode main Costas pulses. A performance comparison of the proposed approaches is supplied. Such waveforms could also be of interest for applications where waveform diversity is desired.

Introduction

Pulse compression is a technique used in radar systems to increase both range resolution and signal to noise ratio. It consists in modulating the transmitted pulse and then correlating the received signal with a replica of the transmitted pulse [START_REF] Skolnik | Radar Handbook, Third Edition[END_REF]. Frequency and phase modulated waveforms are frequently used in pulse compression [START_REF] Levanon | Radar signals[END_REF]. Time-frequency codes are a variant of frequency modulated signals that show high time-frequency resolution. Examples of such codes are Costas codes or Costas arrays [START_REF] Costas | A study of a class of detection waveforms having nearly ideal range ;doppler ambiguity properties[END_REF]. They were introduced as a means of improving performance of radar and sonar systems. Indeed, Costas codes have near ideal range and Doppler sidelobe behavior, providing both unambiguous doppler and range information.

A Costas code consists of an M × M time-frequency code with a randomlike evolution. An example, for M=8, is supplied in Fig 1 . The main feature of these codes is that a single frequency f m , used only once, is emitted in each time interval t p . These codes are designed so as to minimize coincidences between the signal and its time and frequency shifted copies. This property can be evaluated through the ambiguity function AF |χ(τ, ν)|. Ideally, it should be a 2D dirac of maximized energy in (τ, ν) = (0, 0) and zero elsewhere. The properties of Costas codes allow better approximation of the ideal ambiguity function AF, as shown in Fig 2, where the AF of the Costas code [START_REF] Levanon | Radar signals[END_REF][START_REF] Golomb | The status of Costas array construction[END_REF][START_REF] Costas | A study of a class of detection waveforms having nearly ideal range ;doppler ambiguity properties[END_REF][START_REF] Freedman | Staggered Costas signals[END_REF][START_REF] Drakakis | A review of Costas arrays[END_REF][START_REF] Beard | Costas array generation and search methodology[END_REF][START_REF] Skolnik | Radar Handbook, Third Edition[END_REF][START_REF] Golomb | Constructions and properties of Costas arrays[END_REF] is plotted. The thumbtack behavior can be clearly seen and all sidelobes have amplitude 1/M , except a few close to the origin, which are a little bit higher. The literature proposes algorithms for building such codes [START_REF] Golomb | Constructions and properties of Costas arrays[END_REF], [START_REF] Beard | Costas array generation and search methodology[END_REF], [START_REF] Golomb | The status of Costas array construction[END_REF]. A more general approach consists in searching among all possible sequences of length M (all M! sequences), those which follow Costas properties by using the difference matrix method [START_REF] Drakakis | A review of Costas arrays[END_REF]. In this paper, we are interested in studying modified versions of Costas signals. In the literature, some works focused on the modification of those signals in order to improve peak to sidelobe ratio (PSLR) in the autocorrelation function (ACF). A train of staggered Costas signals was proposed in [START_REF] Freedman | Staggered Costas signals[END_REF], in order to allow AF improvement. In the same spirit, an overlaying of orthogonal codes in a train of Costas pulses was proposed in [START_REF] Levanon | Orthogonal train of modified Costas pulses[END_REF]. In [START_REF] Yang | Design and analysis of Costas/PSK RF stealth signal waveform[END_REF], [START_REF] Pace | Costas CW frequency hopping radar waveform: peak sidelobe improvement using Golay complementary sequences[END_REF], [START_REF] Lemieux | Analysis of FSK/PSK modulated radar signals using Costas arrays and complementary Welti codes[END_REF], pulse modulation of Costas frequency hops was introduced, using phase codes and complementary sequences. However, few works study Costas signals beyond the orthogonality condition which requires that frequency separation ∆f should be at most equal to the inverse of pulse duration t p (i.e. t p ∆f = 1). In [START_REF] Levanon | Modified Costas signal[END_REF],

frequency modulation was introduced to perform modification of Costas codes when frequency separation is increased ∆f > 1 tp . This widening is interesting since the overall time-bandwidth product could be achieved without increasing the Costas code size. In this case, the frequency modulation of Costas pulses allows the lowering of autocorrelation function grating lobes that appear when the frequency separation is increased. The work in this paper fits in this specific theme, which is waveform design based on modified Costas signals. Novel modified Costas designs are proposed where the ∆f > 1 tp case, which offers more flexibility to design such signals, is considered. The synthesized waveforms are based on Costas codes combined to pulse waveforms of different natures.

They achieve good autocorrelation properties with lowered grating lobes, when design conditions are respected. The originality of the proposed approaches lies in the diversity introduced by using pulse waveforms of different natures. This can be useful, for instance, for multiple radars systems or multi-user purposes.

The paper is organized as follows. In section II, construction constraints for Costas codes and the modification principle are introduced. The state-of-the-art method for Costas signals modification using Linear Frequency Modulated LFM pulses is also presented. In section III, designs based on different pulse waveforms are proposed. For each case, the relevant parameters that allow a minimum sidelobe combination are defined. In section IV, a method for lowering the first Costas sidelobe is proposed based on previous results. In section V, a performance comparison of the previous approaches is proposed.

Costas signal modification 2.1 Orthogonality constraint and modification principle

The complex envelope x(t) of the Costas code [a m ] m=0:M -1 is given by [START_REF] Levanon | Radar signals[END_REF]:

x(t) = M -1 m=0 u m (t -mt p ) (1)
where

u m (t) = e (j2πfmt) × Rect( t t p )
and Rect(

t t p ) =      1 if 0 ≤ t ≤ t p 0 elsewhere
The design condition ∆f = 1 tp forces signal frequencies f m (m = 0 : M -1) to be multiples of 1 tp that is, f m = am tp . This condition guarantees the orthogonality among Costas pulses and hence low AF sidelobes. However, the orthogonality condition stated before is sometimes restrictive. For example, in some applications, signal duration is limited by radar measurement ambiguities and the bandwidth must be swept in minimum time. If a quick scan is needed, ∆f must be increased beyond the inverse of pulse duration t p , namely ∆f > 1 tp . The frequency in the m th pulse f m changes according to the new t p ∆f product: f m = a m ∆f = am tp × t p ∆f . However, increasing the frequency spacing while keeping rectangular pulses is not convenient. From a spectral point of view, the frequency spacing is increased but occupied by waveforms that do not fill all this spacing. Furthermore, as the orthogonality condition is not satisfied, grating lobes will appear around the mainlobe of the ACF. It was shown in [START_REF] Levanon | Modified Costas signal[END_REF] that around the mainlobe (|τ | < t p ), the ACF R( τ tp ) is roughly the product of R 1 ( τ tp ) the ACF of the rectangular pulse that constitutes the Costas hops and R 2 (τ ) the grating lobes distribution function. Hence,for |τ | < t p :

R( τ t p ) ≈ R 1 ( τ t p ) . R 2 ( τ t p ) where R 2 ( τ t p ) = sin(M πt p ∆f τ tp ) sin(πt p ∆f τ tp ) . (2)
Grating lobes are caused by maxima of R 2 (τ ) that appear at τ (k) = k tp∆f t p where k = 0, ±1, ±2 • • • ± floor(t p ∆f ). In the case of rectangular pulses, R 1 ( τ tp ) has a triangular shape and R( τ tp ) shows equally spaced, linearly decreasing grating lobes, as shown in Fig 3, for the Costas code [START_REF] Levanon | Radar signals[END_REF][START_REF] Golomb | The status of Costas array construction[END_REF][START_REF] Costas | A study of a class of detection waveforms having nearly ideal range ;doppler ambiguity properties[END_REF][START_REF] Freedman | Staggered Costas signals[END_REF][START_REF] Drakakis | A review of Costas arrays[END_REF][START_REF] Beard | Costas array generation and search methodology[END_REF][START_REF] Skolnik | Radar Handbook, Third Edition[END_REF][START_REF] Golomb | Constructions and properties of Costas arrays[END_REF] and ∆f = 5 tp . Based on the above considerations, a change in Costas pulse waveforms must be operated in order to shape pulse ACF R 1 ( τ tp ) in such a way that, when combined with R 2 ( τ tp ), grating lobes are lowered. The authors of [START_REF] Levanon | Modified Costas signal[END_REF] proposed nullifying them using Linear Frequency Modulated LFM pulses instead of rectangular ones. In Costas signals is then given by:

x(t) = M -1 m=0 u m (t -mt p ) (3)
where Basic Costas signals will be the special case A(t) = Rect( t tp ). The discussed pulse waveforms will be detailed in their dedicated parts. The design principle is based on ACF optimization by builiding combinations that guarantee grating lobe lowering. We extend the study to all the delay-Doppler behaviour by studying the AF of modified Costas signals χ(τ, ν). It is derived using results on AF of basic Costas signals, which is given by [START_REF] Costas | A study of a class of detection waveforms having nearly ideal range ;doppler ambiguity properties[END_REF]:

u m (t) = e (j2πfmt) × A(t)
χ(τ, ν) = +∞ -∞ x * (t)x(t -τ )e j2πνt dt χ(τ, ν) = M -1 m=0 e +j2πmνtp [Φmm(τ, ν) + M -1 n=0,n =m Φmn(τ -(m -n)tp, ν)] (4)
where Φ mn (τ, ν) is defined in |τ | ≤ t p by:

Φmn(τ, ν) = +∞ -∞ u * m (t)un(t -τ )e j2πνt dt, |τ | ≤ tp (5)
For modified Costas signals, u m (t) = A(t) × e (j2πfmt) . Hence, for |τ | ≤ t p , Φ mn (τ, ν) is given by:

Φmn(τ, ν) = +∞ -∞ A * (t)e -j2πfm t A(t -τ )e j2πfn(t-τ ) e j2πνt dt Φmn(τ, ν) = e -j2πfnτ +∞ -∞ A * (t)A(t -τ )e j2π(ν+(fn-fm ))t dt Φmn(τ, ν) = e -j2πfnτ χ A (τ, ν + (fn -fm)), |τ | ≤ tp (6)
where χ A (τ, ν) is the AF of the pulse waveform. Replacing Eq 6 in Eq 4, we find the overall AF of modified signals:

χ(τ, ν) = M -1 m=0 e +j2πmνtp χ A (τ, ν)e -j2πfm τ + M -1 n=0,n =m χ A (τ -(m -n)tp, ν + (fn -fm)) e -j2πfn (τ -(m-n)tp ) (7)
In [START_REF] Costas | A study of a class of detection waveforms having nearly ideal range ;doppler ambiguity properties[END_REF], it was stated that around the mainlobe |τ | ≤ t p , the AF mainly depends on the first term of the sum in Eq 7. Hence, the overall AF is related to the pulse waveform AF χ A (τ, ν) as follows:

χ(τ, ν) ≈ χ A (τ, ν) M -1 m=0 e +j2π(mνtp -fm τ ) , |τ | ≤ tp (8)
According to Eq 8, we can establish that the pulse wavefrom AF has a capital contribution to shape the AF of modified signals.

This can be seen more specifically in the zero-Doppler and zero-delay cuts. Noting that

M -1 m=0 e jαm = sin( M α 2 ) sin( α
2 ) e j(M -1) α 2 and f m = a m ∆f , we get the zero-Doppler cut (autocorrelation function ACF) magnitude |R( τ tp )| = |χ(τ, 0)| in Eq 2 according to pulse ACF |R 1 ( τ tp )| = |χ A (τ, 0)|. Note also that the frequency order a m does not affect the zero-Doppler cut in the mainlobe region. Similarly, the zero delay cut is also derived:

|χ(τ, 0)| ≈ |χ A (τ, 0)| sin(M πτ ∆f ) sin(πτ ∆f ) , |τ | ≤ t p |χ(0, ν)| ≈ |χ A (0, ν)| sin(M πνt p ) sin(πνt p ) , |τ | ≤ t p (9)
The lowering of grating lobes results from the relative shapes R 1 ( τ tp ) and R 2 ( τ tp ). In order to achieve the grating lobes mitigation with minimum bandwidth consumption, 3dB bandwidth B p of pulse waveforms must also be considered. To avoid several periods of R 2 ( τ tp ) falling into the main lobe of R 1 ( τ tp ), the first grating lobe position (τ = 1 ∆f ) should be farther than the first null of R 1 ( τ tp )

τ f irst-null that is related to the inverse of pulse waveform bandwidth B p :

1 B p ≤ 1 ∆f ⇒ t p B p ≥ t p ∆f. (10)
In addition to the first grating lobes elimination, this condition also guarantees a good spectral occupancy, since the pulse waveform bandwidth is at least equal to the frequency spacing. In practice, it is preferable to choose B p that fulfills Eq 10 while guaranteeing overlap ratio Bp ∆f as little as possible. Before introducing designed waveforms based on these considerations, we recall, in the next section, the state-of-the-art method for Costas codes modification that involves LFM waveforms.

Costas signals with LFM pulses

A conventional method for grating lobes suppression in Costas

t p ∆f = 4m -n 2m -n and t p B p = (4m -n) 2 2(2m -n) where m, n ∈ N (11)
In Fig 5, an example of Costas signal with size M = 8 coded with LFM pulses is shown, when (t p ∆f, t p B p ) is equal to (7, 24.5).

Grating lobes are significantly lowered by the LFM ACF. Since grating lobes positions do not depend on Costas code size M , the designs conditions are still valid for other Costas sizes.

, n) = (3, 5) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|
Relationships in Eq 11 are derived to position the first two grating lobes at two nulls of Pulse ACF [START_REF] Levanon | Nullifying ACF grating lobes in stepped-frequency train of LFM pulses[END_REF]. However, for fixed t p ∆f , this sometimes gives rise to very high values of t p B p . For example, when t p ∆f = 6, the minimmum value for t p B p is 36:

(m, n) = [START_REF] Beard | Costas array generation and search methodology[END_REF][START_REF] Freedman | Staggered Costas signals[END_REF]. This causes high bandwidth overflow (

Bp ∆f = 6).
It is proposed to alleviate the constraint by just coinciding the first grating lobe with a null of the LFM's ACF, leading thus to πB p k ∆f (1 -k tp∆f ) = mπ [START_REF] Levanon | Nullifying ACF grating lobes in stepped-frequency train of LFM pulses[END_REF] where k is the grating lobe order and m the LFM's ACF null order. Hence, when the m th null of the LFM's ACF coincides with the first grating lobe (k = 1), we have:

t p B p = m (t p ∆f ) 2 (t p ∆f -1) , m ∈ N (12)
With this approach, the other grating lobes are also lowered, to some extent. Bandwidth overflow values are significantly lowered:

in the previous example t p ∆f = 6, the minimal required t p B p is now equal to t p B p = 7.2, and Bp ∆f = 1.2. Some valid configurations of the Costas-LFM combinations, using Eq 12 are presented in Table 1. The choice of m allows us to reconcile lowered grating lobes and limited bandwidth overflow. This design is based on well-known and widely-used LFM chirps. It is well adapted for a single user context but it shows limitations for multi-user applications. Indeed, when multi-user waveforms are needed, the autocorrelation properties must remain acceptable, while providing the lowest possible cross-correlations. This can be achieved by coding differently Costas pulses among different users. Hence, we are interested in other waveforms that introduce more diversity and a large choice among Costas pulse waveforms, thus providing more flexibility for multi-user adaptation.

3 Proposed designs

Costas signals with phase coded pulses

In [START_REF] Touati | Phase coded Costas signals for ambiguity function improvement and grating lobes suppression[END_REF], we have shown that coding pulses, with codes that have good autocorrelation properties, significantly lowers the grating lobes level. The complex envelope of pulse waveforms will be:

A(t) = L-1 l=0 q l Rect( t-ltc tc )
where t c = tp L and [q l ] l=0:L-1 is the code sequence. For a phase code t p B p = L and the condition in Eq 10 will depend on the length of the pulse phase code:

L ≥ t p ∆f. (13)
Either binary phase codes, in which the phase elements are restricted to 0 or π, or polyphase codes, in which there is no restriction on phase increments, can be considered. Probably the most famous family of binary phase codes are Barker codes [START_REF] Borwein | Irreducible polynomials and Barker sequences[END_REF]. Several wellknown polyphase codes are derived from the phase history of LFM pulses. Examples of such codes are Frank codes, P1 and P2 codes, which are only available when L = k 2 [START_REF] Lewis | A new class of polyphase pulse compression codes and techniques[END_REF] and Zadoff-chu codes, P3 and P4 codes, which are defined for any arbitrary length L [START_REF] Lewis | Linear frequency modulation derived polyphase pulse compression codes[END_REF].

For example, a P3 phase code of length L is given by [q l = e φ l ] l=0:L-1 where φ l = 2π L (l) 2

2 . An example of a phase coded Costas signal is provided in Fig 6 for a P3 code of length L = 4. In this case, the ACF nulls of the P3 code of length L = 4 coincide exactly with grating lobes positions and eliminate them. This becomes less obvious when L and t p ∆f are increased. Then, we try to coincide a maximum of nulls with grating lobes positions. Some parameters of valid configurations of the Costas-P4 combination, which were validated by simulations, are presented in Table 2. Parameters are chosen so as to achieve a Peak to Sidelobe Ratio PSLR of 20 dB ±, in the mainlobe region |τ | ≤ t p while keeping the bandwidth overflow B ∆f ≤ 5. This reflects a significant lowering of grating lobes and could be convenient for many radar applications. In Table 2, P4 polyphase codes are used, since there is no restriction for their length (P4 code of length L is given by [q l = e φ l ] l=0:L-1 where φ l = 2π L (l) (l-L) 2 ) and they show good autocorrelation properties that are needed to achieve targeted PSLR. We note that, for a fixed t p ∆f , the longer the code is, the better the lowering of grating lobes will be. This is due to the fact that, for most phase codes, the level of autocorrelation sidelobes decreases when their length is increased, which leads to a better lowering of grating lobes. However, such an increase in phase codes length could be more sophisticated for generation and bandwidth greedy.

(t p ∆f, L) = (2, 4) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|
To design such signals, a large choice of phase codes is available in the literature. This gives more flexibility and diversity for an adaptation to a multi-user context by assigning a unique pulse code to each user.

Costas signals with spectrally balanced signals

Let us now focus on another diversity criterion to select codes. In a multi-user radar context, orthogonality of codes could be desirable in order to better distinguish between users and limit the interference. Hadamard codes are orthogonal codes intensely studied in coding theory. They are also much used in communication systems, mainly in CDMA systems [START_REF] Amadei | On the assignment of Walsh and quasi-orthogonal codes in a multicarrier DS-CDMA system with multiple classes of users[END_REF]. Hadamard codes of length N are a family of N orthogonal binary codes with values +1 and -1. Hadamard codes of lengths N = 2 n are easily built recursively:

for n = 1, the code family is given by the lines (or columns) of matrix

H 2 = [1, -1; 1, 1]. Then for greater n, H n = H 2 ⊗ H n-1 ,
where ⊗ denotes the Kronecker product [START_REF] Ling | Coding Theory, A First Course[END_REF]. However, these codes have poor correlation properties, making them poorly adapted to eliminate grating lobes. We are interested in a modified version of Walsh-Hadamard families where spectral efficiency and correlation properties are largely improved compared to those of original codes. In the following, the use of these codes in Costas pulses is proposed.

Spectrally balanced Walsh-Hadamard codes

In [START_REF] Chonavel | Orthogonal signals with jointly balanced spectra: Application to CDMA transmissions[END_REF], the author proposes a balancing algorithm that transforms a family of orthogonal signals into another orthogonal family that spans the same subspace, so that the spectra of the signals in the new basis have very close spectra. This is achieved by iteratively balancing the energy of the basis signals in a set of prescribed subbands that span the whole signal bandwidth. To this end, a procedure that optimizes the energy balancing criterion is employed. It amounts to successively applying elementary rotations among the entries of the matrix associated to the discretized basis waveforms. An example of a result of this algorithm, when it is applied to 8 × 8

Walsh-Hadamard codes, is provided in Fig 7. The low sidelobe level leads to a considerable lowering of grating lobes level. However, the limiting factor of spectrally balanced Walsh-Hadamard codes is that they can only be generated for lengths that are powers of 2 (L = 2 k ). This restricts significantly the choice among phase codes and hence the t p B p values. A more general approach is to use continuous sequences that can be generated for any bandwidth B p . Good candidates for this approach are Slepian sequences. ACF amplitude at their positions. Hence, it is preferable to use higher t p B p product for Slepian waveforms to guarantee low Slepian ACF sidelobes and good grating lobes elimination. Then, to remain in accordance with the alignment of the first null with first grating lobe position, t p B p must be chosen as a multiple of t p ∆f : 3. For a fixed t p ∆f , we note that the sidelobe level decreases when the sequence bandwidth is increased . This must be considered to select the best sidelobe level-bandwidth compromise. However, spectrally balanced Slepian sequences may show different ACF sidelobes from one sequence to another and this must also be taken into account when designing such signals. When compared to spectrally balanced Hadamard codes, Slepian sequences can lead to less efficient grating lobes lowering, due to a higher ACF sidelobes. However, they offer a large choice of sequences for any given t p B p .

t p B p = k × t p ∆f, k = 1, 2 . . . (14)
N 2 × t s ∆f s ≥ t b ∆f. ( 15
)
When orthogonality condition of secondary Costas is not respected, namely t s ∆f s ≥ 1, ACF of secondary Costas signal R 1 ( τ tp ) will also show its own grating lobes at τ s (k 1 ) = k1 ts∆fs t s with k 1 = 0, ±1 • • • ± floor(t s ∆f s ). Hence, the design should consider this to avoid superposition of main and secondary Costas grating lobes,τ (k) and τ s (k). In other terms, all positions of grating lobes should be different for all possible (k, k 1 ) pairs, except at the origin (main lobe): 4. We note that, when t s ∆f s ≤ 1, for a fixed secondary Costas size N , there are no secondary Costas grating lobes and the design is easier. However, when t p ∆f increases, t s ∆f s must also be increased (for a fixed N ), leading to the appearance of secondary grating lobes. The secondary Costas size N could also be varied in order to find the adequate t p B p = N 2 × t s ∆f s . Such a design is original, since it combines waveforms of the same nature and the many variables of the system allow for better flexibility to adapt it.

τ (k) = τ s (k 1 ) ⇒ k t p ∆f t p = k 1 t s ∆f s t s (16

First sidelobe consideration

the previous sections, we focused on performances in terms of grating lobes lowering. Hence, we did not consider the first sidelobe, which is relatively high. This is a common problem in the standard LFM waveforms and Costas signals where this lobe is approximately -13.7dB below the mainlobe. For some radar applications, this may not be sufficient and it becomes annoying

τ local (k) = (2k+1)
2×M ×tp∆f t p wherek = ±1, ±2 . . . . Especially, if the first sidelobe (k = 1) has to be eliminated, the condition in Eq 10 becomes:

τ f irst-null ≤ τ local (1) 1 B p ≤ 3 2 × M × t p ∆f t p t b B p ≥ 2 3 × M × t p ∆f. (17)
In the case of LFM pulse waveforms, the first null of the LFM can be exactly on the first sidelobe by setting πB p 3 2×M ∆f (1 -3 2×M ×tp∆f ) = π. Then:

t p B p = 4(M × t p ∆f ) 2 3(2 × M × t p ∆f -3) (18)
We show examples of modified Costas signals with respect to these conditions in Fig 14,[START_REF] Touati | Phase coded Costas signals for ambiguity function improvement and grating lobes suppression[END_REF][START_REF] Borwein | Irreducible polynomials and Barker sequences[END_REF] and 17 for all previous designs. This case is bandwidth greedy since t p B p is increased in order to eliminate the first sidelobe. The first sidelobe is considerably lowered, while the other sidelobes level is more or less unchanged.

Performance comparison

The

BT modif ied Costas ≈ M ((M -1) × t p ∆f + t p B p ) (19)
Hence, modified versions introduce flexibility by adding other degrees of freedom to manipulate the time bandwidth product, which are t p ∆f and t p B p . Furthermore, the target BT product could be achieved without increasing the Costas codes size as in

Bandwidth consideration

How are the designs bandwidth greedy for a fixed target PSLR level?. Performances of proposed designs are discussed for a PSLR of 20 dB ± 2 dB, in the mainlobe region |τ | ≤ t p which reflects a significant lowering of grating lobes. Hence, we propose to calculate the mean of values of t p B p in each proposed design, needed to achieve this PSLR. The max PSLR is also reported. The results are shown in Table 5. [START_REF] Lewis | Linear frequency modulation derived polyphase pulse compression codes[END_REF]. However, the advantage of the double coding is the flexibility for an adaptation to a multi-user context, while this is less evident for LFM pulses. Indeed, the number of LFM pulses is limited to one for fixed t p and B p (or two if we consider an up and a down chirp). For some other families of pulse waveforms such as phase codes, Costas codes or Slepian sequences, several waveforms of the family could be generated for a fixed t p and B p . We also note that for all designs, the PSLR can be about 3dB higher than the mean PSLR in the mainlobe region. In what follows, the ambiguity function is considered to discuss the sidelobe behavior and level out of the mainlobe region of the ACF. As can be seen on the ACF (zero Doppler cut), the sidelobe level for Costas-LFM signal is the lowest in the closest region to the mainlobe. However, the volume of the ACF is spread out to higher delays |τ | ≥ t p where it shows higher sidelobes compared to other modified signals that show sidelobes level more regularly distributed along the delay axis. Furthermore, most significant Doppler sidelobes are concentrated around the zero delay axis, for all modified signals. Indeed, the zero delay cut of the AF χ(0, ν), given in Eq 9, depends on zero delay cut of the pulse waveform χ A (0, ν) and a function that shows maxima at multiples of 1 tp that can be seen as Doppler grating lobes. χ A (0, ν) is by definition, the Fourier transform of the squared magnitude of the pulse waveform A(t). Hence, χ(0, ν) depends heavily on amplitude properties of pulse waveforms. For constant amplitude signals it will be a sinc function and these lobes are significantly lowered. However, in the case of balanced signals, it will be different from one code (or a sequence) to another because of different amplitude properties. To avoid high Doppler sidelobes due to Doppler grating lobes, in this case, χ A (0, ν) the radar system should be designed in such a way that maximum detectable Doppler frequency remains smaller than the first Doppler grating lobe position, namely ν max < 1 tp .

Ambiguity Function

Delay-Doppler estimation accuracy

In this section, we evaluate delay-Doppler estimation accuracy using the Cramer-Rao Bound (CRB). Indeed, the CRB is a well known tool to get an indication about best achievable performance of a waveform in terms of delay-Doppler estimation, independent of the filtering algorithm. In the following, CRBs of modified signals are calculated and compared based on AF approximations. Indeed, the CRB is derived from the Fisher Inverse Matrix FIM J -1 M which is directly linked to the second derivatives of the squared magnitude of the ambiguity function Θ(τ, ν) = |χ(τ, ν)| 2 at (τ, ν) = (0, 0) [START_REF] Gini | Waveform Design and Diversity for Advanced Radar Systems[END_REF].

CRB(τ ) = [J -1 M (τ, ν)] (1,1) (20)
CRB(ν) = [J -1 M (τ, ν)] (2,2) (21)
where

J M (τ, ν) = -2SNR    ∂ 2 Θ(τ,ν) ∂τ 2 ∂ 2 Θ(τ,ν) ∂τ ∂ν ∂ 2 Θ(τ,ν) ∂ν∂τ ∂ 2 Θ(τ,ν) ∂ν 2    τ =0,ν=0 (22)
When analytical expressions of χ A (τ, ν) in Eq 8, CRBs can be calculated directly derived. Analytical expressions are available for LFM waveforms and Costas codes but not for phase codes or Slepian sequences. Some approximations can be done to approximate the AF of such signals around the mainlobe which is sufficient to calculate the CRB. After some calculations, AF second derivatives can be derived for each signal. Results are shown in Table 6 ). We also propose to compare Range error bounds with matched filter MF results from simulations. Indeed, the MF is the maximum likelihood detection algorithm and its performances approach the CRBs for high SN Rs. However, the error standard deviation in velocity estimation is relatively high and is not sufficient in practice for most radar applications. Indeed, this is due to the fact that only one signal is used here and the mainlobe is not sharp enough in the Doppler axis.

In practice, a train of identical signals is used to increase signal duration which will lead to a sharpening of the mainlobe and hence reduced estimated Doppler errors.

Spectral properties

To compare spectral properties , we show example of spectra of discussed signals in polyphase codes using a continuous phase modulation CPM in order to achieve spectral containment, could be used [START_REF] Blunt | CPM-based radar waveforms for efficiently bandlimiting a transmitted spectrum[END_REF].

Complexity analysis

A typical hardware architecture to generate proposed signals will be similar to an FHSS Frequency Hopping Spread Spectrum communication system, namely composed of a frequency hop generator involving a frequency synthesizer to select the Costas code a m , (m = 0 : M -1) and generate modulated hopping waves at the frequencies f m and a pulse waveform generator to generate the pulse waveforms A(t) (LFM, phase code, Slepian sequences or Costas code). The outputs of the pulse waveform generator and frequency hop generator are then fed to a modulator where the modulation is done by multiplying the pulse waveform and the modulated hopping waves. Hence, the complexity differences will occur at the waveform generator depending on the pulse nature. The analog generation of frequency modulated pulses, namely LFM and Costas pulses, is the easiest and could be achieved using a voltage-controlled oscillator VCO. However, digital approaches become tempting because of the substantial progress in the computational speed and the size of memory units. Hence, such signals could be generated digitally using Direct Digital Synthesis (DDS) units by supplying the needed frequency evolution at its input which will be linear for LFM pulses, and stepped for Costas.

The complexity of phase codes generation depends, for its part, on the phase code nature. Polyphase codes are moderately more difficult to generate and need sophisticated units, such as DDS. For balanced Slepian sequences, the limiting factor is that they are not governed by a specific equation. Hence, the generation could be done exclusively using memory units to store directly those sequences instead of generating them each time.

Conclusion

This between designs is also proposed to highlight the benefits and disadvantages of each approach. In future works, we will study another advantage, which is the flexibility of proposed designs to an adaptation to multi-user contexts that can help several radars to operate in close proximity.

Figure 1 :

 1 Figure 1: Example of a Costas code of size M=8: [2, 6, 3, 8, 7, 5, 1, 4]

Figure 2 :

 2 Figure 2: Ambiguity function of the Costas code [2,6,3,8,7,5,1,4].

1 -

 1 this paper, alternative pulse waveforms with optimized parameters are proposed. The principle of modification of Costas signals is shown in Fig 4 where rectangular pulses are replaced by other pulse waveforms with complex envelope A(t) of duration t p and a 3dB bandwidth B p . Hence, each frequency hop f m is modulated by the novel pulse waveform A(t). The complex enveloppe of modified-

Figure 3 :

 3 Figure 3: ACF of the Costas code [2,6,3,8,7,5,1,4] in |τ | < t p , for t p ∆f = 5 (Grating lobes), top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )| bottom: |R( τ tp )|

Figure 4 :

 4 Figure 4: Costas signal modification principle.

Figure 5 :

 5 Figure 5: ACF of Costas-LFM signal with (t p ∆f, t p B p ) = (7, 24.5), (m, n) = (3, 5) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

Figure 6 :

 6 Figure 6: ACF of Phase coded Costas signal using P3 code with (t p ∆f, L) = (2, 4) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

Figure 7 :

 7 Figure 7: (a),(b) Walsh-Hadamard codes family L = 8 and corresponding spectra (c),(d) spectrally balanced Walsh-Hadamard codes and corresponding spectra

Figure 8 :Figure 9 :

 89 Figure 8: (a) ACFs of Walsh-Hadamard codes family 8 × 8 (b) ACFs of spectrally balanced Walsh-Hadamard codes 8 × 8

Figure 10 :Figure 11 :

 1011 Figure 10: (a),(b)Slepian sequences family and corresponding spectra (c),(d) spectrally balanced Slepian sequences and corresponding spectra

Figure 12 :

 12 Figure 12: ACF of coded Costas signal using Slepian sequences with (t p ∆f, t p B p ) = (5, 10) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

)Figure 13 :

 13 Figure 13: ACF of doubly coded Costas signal with (M, N, t p ∆f, t s ∆f s ) = (8, 4, 5, 0.78) top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

  overall bandwidth of the modified Costas signals is approximately B modif ied Costas ≈ (M -1)∆f +B p instead of B classic Costas ≈ M ∆f for classic Costas signals. Hence, the time bandwidth product BT changes from BT classic Costas = M 2 to:

Figure 14 :Figure 15 :

 1415 Figure 14: ACF of Costas-LFM signal with (t p ∆f, t p B p ) = (3, 17.5), first sidelobe elimination top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

Figure 16 :Figure 17 :

 1617 Figure 16: ACF of Costas-Slepian signal with (t p ∆f, t p B p ) = (3, 18), first sidelobe elimination top: continuous |R 1 ( τ tp )|, dashed |R 2 ( τ tp )|, bottom: |R( τ tp )|

1 (M - 1 )

 11 Theoretical delay and Doppler resolutions are inversely related to signal bandwidth and signal duration respectively: ∆τ = 1 B = ∆f +Bp and ∆ν = 1 T = 1 M tp . The AF allows us to verify this by evaluating the mainlobe Doppler and delay cuts widths. In the following, examples of AFs of modified Costas signals are proposed to give an overview about sidelobe behavior in all the delay-Doppler plane and not only in the ACF around the mainlobe, as previously done. Results are presented in Fig 18, Fig 19, Fig 20 and Fig 21.

Figure 18 :Figure 19 :Figure 20 :Figure 21 :

 18192021 Figure 18: AF of the Costas-LFM signal with (t p ∆f, t p B p ) = (3, 9)

  for signals with bandwidth B = 100M Hz and duration T = 2.4µs generated using a Costas code of size M = 8. Here, CRBs are evaluated for a Signal to Noise Ratio SNR of 15 dB which is typical for radar applications. Standard deviations of the error on range R and velocity V , σ R = c 2 CRB(τ ) and σ V = c 2fc CRB(ν) are also depicted when the central frequency is (f c = 79GHz

Fig 22 ,

 22 Fig 23, Fig 24 and Fig 25. The Costas-Slepian signals show remarkable spectral properties due to the spectrum balancing operated to the sequences. The LFM-Costas signals and doubly coded Costas signals also present good spectral properties with minimal out-of-band spectral sidelobes. However, the instantaneous phase switching due to the use of phase codes in the Costas-P4 signal causes extended spectral sidelobes. In practice, filters are used to cut off the spectrum tail which can slightly modify the final ACF. Other methods that propose implementing

Figure 22 :

 22 Figure 22: Spectrum of the Costas-LFM signal with (t p ∆f, t p B p ) = (4, 11)

Figure 23 :

 23 Figure 23: Spectrum of the Costas-P4 signal with (t p ∆f, t p B p ) = (4, 11)

Figure 24 :

 24 Figure 24: Spectrum of the Costas-Slepian signal with (t p ∆f, t p B p ) = (4, 12)

Figure 25 :

 25 Figure 25: Spectrum of the Doubly coded Costas signal with (t p ∆f, t p B p ) = (4, 12.5)

  work illustrates several pulse compression techniques based on Costas signals. Methods for modifying Costas signals by widening frequency hops and replacing rectangular pulses by other waveforms that fall within this novel frequency spacing are proposed. This modification allow achieving BT product without increasing the size of the code. Furthermore, it leads to a lowering of autocorrelation grating lobes that appear when the frequency spacing is increased. The originality of the work consists in choosing variable waveforms for the Costas pulses, ranging from traditional signals, such as phase codes and Costas codes, to more sophisticated ones, such as spectrally balanced sequences. The nullification of the first sidelobe in the vicinity of the main lobe is discussed. It is shown that it is possible to suppress it as well as grating lobes, by a convenient choice of the bandwidth of the pulse waveform. A comparison

Table 1 :

 1 Example of valid configurations of Costas-LFM combination (Costas code M = 8) m t p ∆f t p B p Bp ∆f m t p ∆f t p B p

	Bp
	∆f

Table 2 :

 2

	Bp
	∆f

Example of valid configurations for phase coded Costas signals (Costas code M = 8 and P 4 phase codes) t p ∆f t p B p (L) Bp ∆f t p ∆f t p B p (L)

Table 3 :

 3 Costas t p is subdivided into N sub-pulses of duration t s to contain the secondary Costas, i.e. t p = t s × N . The bandwidth of the secondary Costas is B p = N × ∆f s and hence t p B p = N 2 × t s ∆f s . Then, the condition in Eq 10 becomes :

	Bp
	∆f

Example of valid configurations of Costas-Slepian combination (Costas code M = 8) t p ∆f t p B p Bp ∆f t p ∆f t p B p In this section, other Costas codes are used to encode the initial Costas pulses. The doubly coded Costas signal consists in replacing rectangular pulses by another Costas signal

[START_REF] Touati | Doubly coded Costas signals for grating lobes mitigation[END_REF]

. Indeed, under some conditions upon secondary Costas parameters, we have shown that such a double Costas coding approach can be efficient in eliminating grating lobes. The secondary Costas pulse duration is t s and frequency spacing is ∆f s . The complex envelope of the Costas pulse is given by: A(t) = L-1 l=0 v s (t -lt s ) where v s (t) = Rect( t ts )e j2πf l t and f l = l∆f s are secondary Costas code elements. The only restriction on this design is that pulse duration of main

Table 4 :

 4 Example of valid configurations of Doubly coded Costas signals: Main Costas code M = 8, secondary Costas code N = 4 t p ∆f t s ∆f s t p B p Bp ∆f t p ∆f t s ∆f s t p B p Hence, it is proposed to extend the study to first sidelobe elimination, by positioning first null of Costas pulse ACF around the first sidelobe of R 2 ( τ tp ) (instead of first grating lobe as for previous designs). High sidelobes are positioned between two successive grating lobes. They appear at local maxima of R 2 ( τ tp ) (maxima of the numerator):

	Bp
	∆f

Table 5 :

 5 Comparison of mean values of t p B p and achieved mean PSLR for the state-of-the-art method and the proposed designs We can see that doubly coded Costas signals require more bandwidth than phase coded Costas signals and Slepian coded Costas signals. These last achieve the fixed PSLR with much less bandwidth. LFM approach cannot be compared very precisely to our approaches in terms of bandwidth consumption, since they do not have the same target PSLR. However, it can be compared to the doubly coded design since it has approximately the same bandwidth consumption. Using LFM pulses achieve a better PSLR (+5dB) than doubly coded Costas signals, with approximately the same t p B p (≈

	Costas pulse nature LFM Phase code Slepian Costas code
	PSLR (dB) (mean) 25.3	20.8	20.6	20.4
	PSLR (dB) (max)	28	22.6	23.1	23.5
	t p B p (mean)	17.9	12.9	9	17.7
	Bp ∆f (mean)	3.07	2.33	1.5	3.31

Table 6 :

 6 CRBs and MF results for different modified Costas signals, when an SN R = 15dB is considered

	Pulse waveform LFM	P4	Slepian	Costas
	(t p ∆f, t p B p )	(3, 9)	(3, 9)	(3, 9)	(3, 8.8)
	CRB(τ )(s 2 )	4.5342 × 10 -19 4.4898 × 10 -19 4.5564 × 10 -19 3.5842 × 10 -19
	σ R (m)	0.1010	0.1005	0.1013	0.0898
	σ M F (m)	0.1259	0.1054	0.1353	0.1258
	CRB(ν)(s -2 )	7.5213 × 10 8	7.0879 × 10 8	7.2497 × 10 8	6.8088 × 10 8
	σ V (m/s)	52.07	50.55	51.12	49.54
	All signals have acceptable error standard deviation in range estimation. This could be verified by matched filter results which

are very close to CRBs for all signals. Costas -P 4 achieve the minimal range error estimation using MF which is about 10.54cm.

Touati, Nadjah; Tatkeu, Charles; Chonavel, Thierry; Rivenq, Atika: Design and performances evaluation of new Costas-based radar waveforms with pulse coding diversity', IET Radar, Sonar and Navigation, 2015, DOI: 10.1049/iet-rsn.2014.0450


Touati Nadjah
email: nadjah.touati@ifsttar.frcharles.tatkeu@ifsttar.fr

Tatkeu Charles

Rivenq Atika

El Hillali Yassin

Chonavel Thierry
email: thierry.chonavel@telecom-bretagne.eu

Pulse compression waveforms design and Cross-ambiguity processing for Radar applications

Keywords: Radar waveforms, Costas signals, Auto-ambiguity function, Cross-ambiguity function, matched filter

Radar waveforms design is an important step to design a hole radar system. In this paper, several waveforms based on modified Costas codes are used. The properties and performances are tested and compared. We propose a detection based on cross-ambiguity calculation for these waveforms. Indeed, such a processing is interesting when delay and Doppler should be estimated. The performances are evaluated in several scenarios in presence of noise and multiple targets. The waveforms are also tested on real experimentation hardware.

I. INTRODUCTION

In radar applications, high range resolution may be obtained with short pulses and hence large bandwidths. However, short pulse radars shows limitations due to the high peak power required to achieve the large pulse energy. On the other hand, a long pulse can have the same bandwidth (resolution) as a short pulse if it is modulated in frequency or phase. Pulse compression is a technique that allows a radar to simultaneously achieve the energy of a long pulse and the resolution of a short pulse by modulating the pulse [START_REF] Skolnik | Radar Handbook[END_REF]. Two most important classes of pulse compression waveforms are frequency modulated signals and phase coded signals [START_REF] Levanon | Radar signals[END_REF]. Costas codes are a sort of frequency hopping codes that achieve the thumbtack auto-ambiguity function AAF needed in radar applications [START_REF] Costas | A study of a class of detection waveforms having nearly ideal range-doppler ambiguity properties[END_REF] thanks to their optimized frequency evolution.

In this paper, waveforms based on modified Costas signals are synthesized. These waveforms offers flexibility to Costas codes by introducing pulse diversity. More specifically, the modification consists in widening the frequency spacing between Costas frequency hops and replacing constant frequency pulses by other pulse compression waveforms. A detection using cross-ambiguity function CAF calculation is operated for proposed signals. Performances are tested in a Radar scenario considering noise and single/multiple target context. The tests of the waveforms on real experimentation Hardware are also shown.

The paper is organized as follows: in section II, the principle and properties of used waveforms, in this work, are discussed. In section III, the principal of target detection using cross-ambiguity calculation is exposed and applied for the above signals in presence of noise and/or multiple targets. Finally in section IV, some real experimentation results are shown.

II. WAVEFORM DESIGN

In this part, we will discuss pulse compression waveforms used in this work. Indeed, we use waveforms based on modified Costas codes. The modification consists in widening frequency separation between Costas frequency hops and replace constant frequency pulses with other waveforms. tp . This condition forces the AAF to have a single centered mainlobe and low sidelobes at all other points. This condition imposes a gradual scan of the bandwidth by a 1 tp step. Hence, it is sometimes interesting to increase the frequency spacing beyond this condition that is setting ∆f > 1 tp . However, increasing the frequency spacing while keeping constant frequency pulses is not convenient. Indeed, when orthogonality condition is not satisfied, grating lobes around the main temporal lobe of the autocorrelation function ACF of the signal will appear. Based upon previous considerations, a change in Costas pulse waveforms is operated in order to allow the lowering of grating lobes. Depending on pulse waveform nature, several modified signals exists: b) Costas-LFM signals: Modified Costas signals where proposed when (∆f > 1 tp ), in [START_REF] Levanon | Modified costas signal[END_REF]. They consists in replacing constant frequency pulses by Linear Frequency Modulated LFM ones. This approach involves placing nulls of single LFM ACF exactly where the grating lobes are located, and thus removes them completely. This takes place under the restrictions of an appropriate choice of the LFM bandwidth B p according to Costas frequency spacing. c) Phase coded Costas signals: Another way to modify Costas codes consists in introducing phase coding in Costas pulses [START_REF] Touati | Phase coded costas signals for ambiguity function improvement and grating lobes suppression[END_REF]. When used phase codes show good autocorrelation properties grating lobes are lowered significantly. Furthermore, to avoid several periods of the signal to fall in the mainlobe the [START_REF]Doubly coded costas signals for grating lobes mitigation[END_REF]. Indeed, under some conditions on secondary Costas parameters: size N , sub-pulse duration t s and frequency spacing ∆f s , such a double Costas coding approach can be efficient in eliminating grating lobes. Indeed, the bandwidth of the secondary Costas B p = N ×∆f s must be at least to the novel spacing i.e. N 2 × t s ∆f s ≥ t b ∆f .

Previous described in are synththised for a fixed bandwidth B = 500M Hz. For modified Costas codes, we must select the pair of compatible (t p ∆f, t p B p ) according to conditions stated in [START_REF] Levanon | Modified costas signal[END_REF], [START_REF] Touati | Phase coded costas signals for ambiguity function improvement and grating lobes suppression[END_REF] and [START_REF]Doubly coded costas signals for grating lobes mitigation[END_REF]. The overall bandwidth of the modified Costas signals is approximatively:

B modif ied Costas ≈ (M - 1)∆f + B p instead of classical Costas signals bandwidth B classical Costas ≈ M ∆f = M
tp . This allow a better flexibility to achieve the needed bandwidth. Indeed, we fix the main Costas code [1, 3, 6, 7, 5, 4, 8, 2] of size M = 8 and calculate the pulse duration t p , frequency spacing ∆f according to the novel t p ∆f product, and the overall signal duration T in each case. Results are shown in Table I, where an LFM chirp, a P4 phase code , a spectrally balanced Hadamard code and a Costas code are used to encode the main Costas pulses.

The auto-ambiguity function AAF represents the time response of a filter matched to given finite energy signal when the signal is received with a delay τ and a Doppler shift ν relative to the nominal values (zeros) expected by the filter [START_REF] Levanon | Radar signals[END_REF]. The AAF is expressed by:

|χ u,u (τ, ν)| = | + inf -inf u(t)u * (t + τ )e (j2πνt) dt| ( 1
)
where u is the complex envelope of the signal. This function allow evaluating performances of Radar waveforms. The ideal one consists in one centered spike and zeros elsewhere. However, in a real-life scenario, the received signal is the sum of multiple echoes delayed and Doppler shifted, corrupted by noise. The cross-ambiguity function CAF must be considered to find the spikes that will be centered in the true delays and Doppler shifts.

III. CROSS-AMBIGUITY FUNCTION

When a signal u(t) is transmitted. The received echo signal will be the sum of several echoes corrupted by noise. It can be expressed by: r(t) = K i=1 α i u(t -τ i )e (j2πνi(t-τi)) + n 0 (t), where n 0 (t) is an additive noise and (τ i , ν i ) are delay and Doppler of the i th target. The CAF |χ u,v (τ, ν)| between the reference signal and the received echo, will be the sum of individual AF's centered in the true delay and Doppler shifts:

χ u,r (τ, ν) = K i=1 α i χ u,u (τ -τ i , ν -ν i )e -j2πνi(τ -τi) + n 1 (t) (2)
where n 1 (t) is the noise at the output of the crossambiguity function. In this section, we will present the application of the cross-ambiguity function CAF in a radar context. We will introduce the parameters of a radar into the CAF and display it. The purpose of this is two-fold: display performances of previous signals in presence of noise and multiple targets. The principle of calculation is presented in Fig 1. We can see that the CAF is almost transparent to the noise effect for SNR higher than -10dB. This is justified by the matched filter since it is the optimal linear filter for maximizing the SNR in the presence of additive noise. The effect start to We note that theoretical ACF and practical CCF are very similar. Indeed, the hardware effect, specially , the antenna differentiation, is not much visible and the CCF is preserved and produce a spike on the correct delay corresponding to the distance between the antenna and the target. The CAF's also show a centered spike in the correct delay and Doppler (τ, ν) ≈ (15.95ns, 0). Also, returns form the several signals are likely similar in the vicinity of the main lobe. This is due to surrounding environment and antenna leaks. Several experimentations must be done, considering moving targets also, in order to better evaluate performances and estimation accuracy.

V. CONCLUSION

In this paper, a detection scheme based on cross-ambiguity calculation is proposed. First, waveforms based on modified Costas codes are synthesized. A study of cross-ambiguity function properties on such waveforms is then proposed. Performances in presence of noise and multiple targets are evaluated. It was seen that proposed signals offer more robustness to noise and multiple targets because of the low sidelobes level. Finally, some experimentation results has been shown in order to validate the waveforms and evaluate their performances in a real-life scenario. This work should be extended to the evaluation of performances in terms of Delay and Doppler estimations. Furthermore, we must focus on more sophisticated scenarios in experimentation that help understand the radar environment constraints.

  a) Standard Costas signals: A Costas signal consists in an M × M time-frequency code with a randomlike ordering [a 1 a 2 ...a M ]. Each frequency pulse consists in a constant frequency signal of duration t p . The frequency hopping ∆f is at least equal to the inverse of pulse duration, ∆f = 1

  ACF's of signals of previous signals are free of grating lobes thanks to valid combinations between main Costas parameters and pulse waveforms parameters as shown in reference works. The AAF's also result in centered spike and low sidelobes distributed in the delay-Doppler plan. The spike have a temporal width of 1/B and Doppler width of 1/T . We note that modified signals show low sidelobes compared to standard Costas signals especially in the vicinity of the main lobe.

Fig. 1 .Fig. 2 .

 12 Fig. 1. Cross-ambiguity function CAF calculation principle between the reference signal and a noisy, time-delayed, frequency-shifted version a) CAF in noisy environment: In this part, we propose to evaluate CAF properties in presence of noise. We propose to vary the SN R = -20, -10, 0 and 10dB. Results are presented in Fig 2, Fig 3, Fig 4, Fig 5, Fig 6. The operational scenario constructed here is when the received signal is corrupted by a delay τ = 16ns and a Doppler ν = 62.5M Hz.

Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 3456 Fig. 3. CAF for a Costas-LFM signal: a) SNR=10dB, b) SNR=0dB, c) SNR=-10dB, d) SNR=-20dB

Fig. 7 .Fig. 9 .Fig. 12 .

 7912 Fig. 7. CAF for a standard Costas signal: a) spaced targets scenario c) close targets scenario

  Fig. 13. a) CAF for a standard Costas signal, b) ACF and CCF for a standard Costas signal, c) CAF for a Costas-Hequi signal, d) ACF and CCF for a Costas-Hequi signal

TABLE I .

 I MODIFIED COSTAS SIGNALS FORA COSTAS SIZE M = 8 AND A TOTAL BANDWIDTH B ≈ 500M Hz The doubly coded Costas signals principle consists in using other Costas codes to encode the principal Costas pulses

		Standard Costas	Costas-LFM	Costas-Hequi	Costas-P4	Doubly coded Costas
	Main Costas size	8	8	8	8	8
	(tp∆f, tpBp)	(1,0)	(6,14.5)	(6,8)	(6,9)	(6,14.4)
	tp(ns)	16	113	100	102	112.8
	∆f (M Hz)	62.5	53.1	60	58.8	53.2
	B(M Hz)	500	500	500	500	500
	T (ns)	128	904	800	816	902

phase code length must be at least equal to the novel spacing i.e. L ≥ t p ∆f d) Doubly coded Costas signals:


T Balan

F Abed-Meraim

Numerical Investigation of the Limit Strains in Sheet Forming Involving Bending

In this work, the finite element method is used to simulate a typical FLD test over tools of different radii. Parameters like the mesh density, element type, numerical determination of the onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit strain for plane strain tension has been determined as a function of the thickness vs. tool radius (t/R) ratio. These simulations confirm that increasing the curvature of the tool increases the value of the limit strains. They also reveal that, as soon as bending becomes important, the practical relevance of the limit strains diminishes -at least with their current definition. The need for new strain localization models is emphasized, together with some of the associated challenges.

Introduction

Forming limits represent a key issue in sheet metal forming. The concept of forming limit diagram (FLD) has proved to be very useful in describing the forming limits of sheet metals in a general and practically relevant manner. Decades of effort have given rise to standardized experimental methods, as well as numerous modelling approaches, for the determination / prediction of the FLDs.

The development e.g. of high strength steels and the associated forming technologies revealed several limitations of the existing formability models. Among other limitations, FLDs were not aimed for application in parts with very small radii, but rather for the study or large panels in areas of reasonable curvature. The influence of curvature was already first pointed out in the early work of Gosh and Hecker [1]. In an attempt to extend the range of application of the FLD, Col [START_REF] Col | FLCs: Past, present and future[END_REF][START_REF] Col | FLCs: are we at a turn?[END_REF] suggested the new concept of FLS -"Forming Limit Surface" (see Fig. 1). While its beneficial effect is largely recognised in the sheet forming press-shops, scarcely any data is available to support the speculations about the quantitative impact of the sheet/die curvature on the FLD. However, such information would be a prerequisite for the development of more realistic formability models and criteria. This work is an attempt to quantitatively investigate the impact of curvature on the FLD. The analysis is restrained to the "FLD 0 " point, corresponding to the plane strain mode. The finite element method is used to simulate typical FLD tests over dies of different radii. Parameters like the mesh density, element type, numerical determination of the onset of strain localization, limit strain definition etc. have been investigated. Finally, the limit strain for plane strain tension has been determined as a function of the thickness vs. die radius ratio.

Definition of the model problem

Due to its well known strain-path dependency, the forming limit diagram is formally determined for proportional (or linear) strain paths -while an infinite variety of non-linear strain-paths are often encountered in practice. Similarly, there is no unique manner to overlap bending to a typical (even linear) in-plane loading of a metal sheet. Obviously, the ratio of these two contributions and its history of application could have an important role on the limit strains. In this work, a particular choice is made that is inspired from the practice of the Nakajima FLD test: metal sheet is supposed clamped on its external contour and it is drawn over a punch with a circular profile. Since the plane strain loading mode is given particular attention, a two-dimensional, plane-strain configuration is considered (see Fig. 2).

Friction is neglected on both contact interfaces; in order to enforce localization on the top of the punch, the die radius is chosen slightly larger than the punch radius. The length of the sheet sample and the die radius have been tuned by means of numerical tests, so that their particular values do not affect the results.

ε 2 t/R 0,0 0,3 0,6 0,9 0,6 0,5 0,4 0,3 0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 ε 2 t/R 0,0 0,3 0,6 0,9 0,6 0,5 0,4 0,3 0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 ε 2 t/R 0,0 0,3 0,6 0,9 0,6 0,5 0,4 0,3 0,2 0,1 0 -0,1 -0,2 -0,3 -0,4 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 t/R

Fig. 1: Forming Limit Surface concept introduced by Col [2].

In order to address the effect of the thickness vs. punch radius ratio, three values have been tested for the punch radius: 50, 25 and 12.5 mm. They correspond to thickness vs. punch radius ratios of 0.06, 0.12 and 0.24. An arbitrary sheet thickness of 3 mm has been chosen but its particular value should not affect the results, provided that the above ratios are respected. Solid finite elements are used for the simulations, in order to describe the strain heterogeneity in the pole region, due to bending and contact. Hybrid, displacement -pressure linear elements have been used with an enhanced strain definition aiming to accommodate incompatible modes due to bending. The simulations are performed with Abaqus/Standard. Fig. 3 gives an outline of the geometry and the finite element mesh. In the zone of interest, a number of ten finite elements in the thickness direction (twenty integration points) are used to correctly describe the strain gradient and to avoid mesh dependency.

Fig. 3:

Finite element mesh at the beginning of the simulation and sample results at the end of the simulation. Top: detail of the mesh at the pole region. Bottom: the in-plane major strain is measured on the top surface of the sheet, along the thick line, starting from the pole.

The strain analysis is made difficult by the presence of a strain gradient through the thickness of the sheet, due to bending. Since the experimental strain measurements are always made on the outer surface of the sheet, the same surface is conventionally used in the present analysis for the quantitative evaluation of strains. In fact, the principal in-plane strain is recorded on the outer surface of the sheet, starting from the pole -as indicated in Fig. 3. For large values of the punch radius, the strain value at the pole measured in this way should correspond to the so-called FLD 0 value -the lowest point of the FLD, laying on the vertical axis. The primary aim of this analysis is to quantify the evolution of this value when the punch radius is diminishing.

The material is considered isotropic (von Mises yield surface); the evolution of the flow stress Y (hardening) is described with a classical Voce law:

0 (1 ) b Y Q e ε σ - ∞ = + - with 0 200 MPa σ = , 400 MPa Q ∞ =
and b = 10. As already underlined, at the current stage the results are not to be directly related to an experimental situation -so the material has also been taken into account in this simplified manner.

Results of the FE simulations

Finite element simulations have been performed for the three values of punch radius. Among other quantities, the major strain has been recorded on the top surface of the sample. Fig. 4 shows a heterogeneous strain distribution corresponding to the zone affected by contact with the punch. Globally, the largest values of strain are observed in a small zone close to the pole -although the maximum is not necessarily located at the pole. The necking process becomes more and more catastrophic and eventually, strain localization occurs. However, this is not an instantaneous phenomenon and the definition of the exact moment of strain localization is not unique -as it is also the case experimentally. Moreover, the numerical results become less and less reliable as the strain localizes -due to pathological mesh sensitivity. Fig. 4 illustrates the evolution of the strain distribution in the zone of interest at several moments in the loading history. The thick line designates a particular simulation increment, when most of the sheet is elastically unloaded suddenly, after what the plastic strain occurs only in small areas in the bent zones (this is especially true when only the outer surface of the sheet is analyzed). The strain localization relevant for sheet metal forming is often considered to occur somewhat later. However, it is clear from Fig. 4 that the subsequent curves can hardly be used for quantitative strain evaluations. Indeed, the evolution of strains becomes strongly mesh dependent when most of the mesh is unloaded. On the other hand, damage should also be considered in the constitutive model if accurate limit strains are searched for at such large values of strain. In contrast, the numerical simulations with different mesh densities have shown almost identical strain distributions at the moment of global elastic unloading of the sample. According to these observations, the sudden elastic unloading of more than 80% of the sample is formally considered as the localization criterion in this study. Again, this criterion is not aimed to mimic the localization criterion adopted experimentally; it mainly allows a consistent quantitative comparison of the results for different punch radii. It is however expected that the trends of this criterion would correspond to the trends of more realistic localization criteria, in terms of t/R sensitivity -and remains representative for applications to sheet metal forming.

The same remarks about the strain distribution stand whatever the value of the die radius, as seen in Fig. 5. In order to compare the results to the ideal situation with no bending, a simple, tool-less tensile test (plane strain) has been simulated using the same initial mesh. The thick lines from all sub-figures in Fig. 5, corresponding to the strain distributions at localization, are overlapped in Fig. 6 (left). In the four cases, the strain distribution can be roughly described by two values: the value at the pole -the so-called FLD 0 value -and the value far from the pole, in the zone unaffected by bending. In real situations, this later zone is the largest one for many real parts.

The values of these two representative strains are summarized in Fig. 6 (right) for the different t/R ratios. Obviously, the two values are almost identical in the case of in-plane (tool-less) straining -due to the particular choice of the localization criterion considered here. When the t/R ratio is increasing, the FLD 0 strain is increasing, thus confirming the intuitive trend from Fig. 1. By contrast, the strain level far from the pole is decreasing with the t/R ratio. Major strain distributions at strain localization. Right: FLD 0 limit strain dependency to the t/R ratio.

Discussion and conclusions

Simple finite element simulations have shown that the impact of bending on the limit strains in plane strain loading can be quantified numerically. This effect is beneficial -strain localization is delayed by bending. However, the strains attained far from the pole are decreasing when the thickness vs. punch radius ratio t/R is increasing. Consequently, the limit strains have a very local meaning and are no longer characteristic of the overall straining of the sheet.

In order to relate the numerical results to experiments, one has to carefully adopt the localization criterion when limit strains are considered attained. The criterion adopted here was the sudden unloading of most of the sample; this choice prevents a too important sensitivity of the results to mesh density. On the other side, this criterion is easy to adopt in finite element simulations -while it would be more difficult to apply for experiments.

The current analysis has several limitations. First of all, the results depend at some extent on the mesh. The mesh density has been tuned by numerical tests to avoid such dependency -however it is far too dense to be used in real stamping applications. Eventually, the limit strains at the pole (on the outer surface of the sheet) have less practical relevance when bending becomes important: they give no information about the underlying strains through the thickness of the sheet -and they are also not representative of the strains beyond the tool radius.

The prediction of strain localization appeals for new localization criteria and models, taking into account the curvature of the sheet. However, few or no localization criteria currently in use allow a straightforward extension to heterogeneous strain fields. Moreover, one can expect that when the strains are increasing, damage should not be neglected in the material modelling. Finally, the mesh dependency remains a major difficulty that must be taken into consideration.

Numerical simulations like the ones performed in this analysis can bring insight about the required features of such new models -and could provide a means of validation. In future work, these simulations will be extended to other straining modes (uniaxial tension, balanced biaxial tension etc.) in an attempt to simulate the forming limit surface (Fig. 1). This purely numerical study will be accompanied by a modelling effort to take curvature into account in strain localization.

Fig. 2 :

 2 Fig. 2: Geometry and loading considered for the numerical simulations.

Fig. 4 :

 4 Fig. 4:Major strain distribution at different stages of the simulation; punch radius 12.5 mm.

Fig. 5 :Fig. 6 :

 56 Fig. 5: Major strain distribution at different stages of the simulations, for different punch radii (from top left to bottom right: no radius; R=50 mm; R=25 mm; R=12.5 mm).


Denis Efimov

Andrey Polyakov

Leonid Fridman

Wilfrid Perruquetti

Jean-Pierre Richard

J.-P Richard

Delayed Sliding Mode Control

Keywords: Sliding mode, Time delay, Input-to-state stability

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

I. INTRODUCTION

The sliding mode control and estimation algorithms became very popular nowadays due to their strong advantages: finite time of convergence and compensation of matched disturbances [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF], [START_REF] Fridman | Sliding Mode Enforcement after[END_REF], [START_REF] Boiko | Discontinuous Control Systems: Frequency-Domain Analysis and Design[END_REF]. Despite of that the sliding mode solutions have also several drawbacks mainly originated by the impossibility of a perfect practical realization of sliding motion. Among these shortages it is necessary to mention the chattering phenomenon, which is a high frequency oscillation of control signal when trajectories stay around the sliding surface. The appearance of chattering may physically destroy actuator and/or degrade the performance of transients [START_REF] Utkin | Sliding Modes in Control Optimization[END_REF], [START_REF] Levant | Chattering analysis[END_REF]. There are several approaches for chattering reduction [START_REF] Fridman | Sliding Mode Enforcement after[END_REF]. One of the most popular deals with the high order sliding mode algorithms [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF], [START_REF] Edwards | Continuous higher order sliding mode control based on adaptive disturbance compensation[END_REF], where the discontinuity, for example sign[y(t)] =    y(t) |y(t)| , y(t) = 0 0, y(t) = 0 , with y(t) ∈ R and |•| is the absolute value, appears not in the control/estimated variable itself but in its derivative of higher order [START_REF] Perruquetti | Sliding Mode Control in Engineering[END_REF], [START_REF] Fridman | Sliding Mode Enforcement after[END_REF]. However, since a discontinuity is present in the system closed loop, it may negatively influence the transients through the corresponding derivative (if this derivative has a physical meaning in the system). That is why frequently for practical realization of sliding mode algorithms different approximations of sign functions are used [START_REF] Ambrosino | Robust model tracking control for a class of nonlinear plant[END_REF], [START_REF] Burton | Continuous approximation of variable structure control[END_REF], like for instance ŝign[y(t)] = y(t) + |y(t)| with some sufficiently small > 0, or see also [START_REF] Canudas-De Wit | Smoothing strategies for high-gain control[END_REF] for interesting general sigmoid and dynamic approximations. Such a control based on approximated sign functions received the name of "continuous" sliding mode [START_REF] Khalil | Nonlinear Systems[END_REF], [START_REF] Oza | Continuous second order sliding mode based robust finite time tracking of a fully actuated biped robot[END_REF], [START_REF] Shtessel | An asymptotic second-order smooth sliding mode control[END_REF]. The main drawback of existent approximations is that a chattering reduction is achieved by a price of quality loss (appearance of static error in the presence of matched disturbances and, consequently, practical stability with an exponential rate of convergence [START_REF] Ambrosino | Robust model tracking control for a class of nonlinear plant[END_REF], [START_REF] Esfandiari | Stability analysis of a continuous implementation of variable structure control[END_REF]). In the present work a development of the sliding mode control is presented that is based on a sign approximation using the time-delay framework, which guarantees the quality preservation (there is no static error in the presence of a matched disturbance and locally the speed of convergence is faster than any exponential).

Usually appearance of a delay in the system leads to performance degradation and complication of stability analysis. However in this work we are going to introduce delay in a proper way into the system, in order to make a modification of the sign function, proving certain performance in the system. The obtained approximation is very simple and can be easily implemented in control/estimation systems based on the sliding mode algorithms:

sign τ (y t ) = y(t) max{|y(t)|, |y(t -τ )|} ,
where y t ∈ C [-τ,0] , y(t) ∈ R is a variable whose sign has to be evaluated at time instant t ∈ R and τ > 0 is a fixed delay (design parameter). Obviously,

sign 0 (y t ) = sign[y(t)].
In this work the new delayed sliding mode control framework will be substantiated, it is based on such an approximation and admits the following advantages:

• exact cancellation of matched disturbances is preserved despite of approximation;

• the control algorithms demonstrate hyper-exponential convergence to the origin;

• chattering reduction with respect to conventional sliding mode control.

The idea of chattering reduction, roughly speaking, is based on the fact that

{φ ∈ C [-τ,0] : sign τ (φ) = 0} ⊂ {φ ∈ C [-τ,0] : sign[φ(0)] = 0}.
The outline of this work is as follows. The preliminary definitions for time-delay systems are given in Section 2. The motivating example of the first order sliding mode control algorithm is considered in Section 3. An extension to high order sliding mode algorithms is presented in Section 4. Several examples are considered for illustration of the obtained results.

II. PRELIMINARIES

Consider a functional differential equation of retarded type [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF]:

dx(t)/dt = f (x t , d(t)), t ≥ 0, (1)
where

x(t) ∈ R n and x t ∈ C [-τ,0] is the state function, x t (s) = x(t + s), -τ ≤ s ≤ 0 (we denote by C [-τ,0] , 0 < τ < +∞ the Banach space of continuous functions φ : [-τ, 0] → R n with the uniform norm ||φ|| = sup -τ ≤ς≤0 |φ(ς)|, where | • | is the standard Euclidean norm); d(t) ∈ R m is an essentially bounded measurable input, i.e. ||d|| ∞ = ess sup t≥0 |d(t)| < +∞; f : C [-τ,0] × R m → R n
is a locally bounded functional, f (0, 0) = 0 where 0(s) = 0 for all -τ ≤ s ≤ 0. The representation (1) includes pointwise or distributed time-delay systems. We consider the system (1) with the initial functional condition

x 0 ∈ C [-τ,0] .
A continuous function σ : R + → R + belongs to class K if it is strictly increasing and σ (0) = 0; it belongs to class K ∞ if it is also unbounded. A continuous function β : R + × R + → R + belongs to class KL if β(•, r) ∈ K and β(r, •) is a decreasing to zero for any fixed r ∈ R + .

A. Discontinuous functional differential equations

The dynamical systems subjected by a time-delay, whose models are described by functional differential equations, find their applications in different areas of science and technology [START_REF]Applications of Time Delay Systems[END_REF]. Analysis of delay influence on the system stability is critical for many natural and human-developed systems [START_REF] Gu | Stability of Time-Delay Systems[END_REF], [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF], [START_REF] Richard | Time-delay systems: an overview of some recent advances and open problems[END_REF]. The problem of stability investigation in time-delay systems is much more complicated than for ordinary differential equations since design of a Lyapunov-Krasovskii functional or a Lyapunov-Razumikhin function is a complex issue.

It is known from the theory of functional differential equations [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF] that under the above assumptions the system (1) with a locally Lipschitz f has a unique solution x(t, x 0 , d) satisfying the initial condition x 0 for the input d(t), which is defined on some finite time interval [-τ, T ) (we will use the notation x(t) to reference x(t, x 0 , d) if the origin of x 0 and d is clear from the context). If function f (φ, d) ≡ f 0 (φ(0), φ(-τ ), d), where f 0 : R 2n+m → R n , and it is discontinuous with respect to φ(0) on a set N ⊂ R n of measure zero only and continuous with respect to φ(-τ ) and d, following [START_REF] Filipov | Differential Equations with Discontinuous Righthand Sides[END_REF], [START_REF] Heemels | Input-to-state stability and interconnections of dicontinuous dynamical systems[END_REF] (see also [START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF], [START_REF] Surkov | On functional-differential equations with discontinuous right-hand side[END_REF] for a general definition and existence conditions of solutions for discontinuous functional differential equations) we will consider its multi-valued extension (define

B ε (x) = {y ∈ R n : |x -y| ≤ ε} as a closed ball of radius ε > 0 around x ∈ R n ): F (φ, d) = ε>0 conv[f 0 {B ε (φ(0)) \ N , φ(-τ ), d}],
which is non-empty, compact, convex and upper semi-continuous [START_REF] Heemels | Input-to-state stability and interconnections of dicontinuous dynamical systems[END_REF] for any d ∈ R m . In particular, the multi-valued extension of sign τ (y t ) can be defined as follows:

sign τ (y t ) =    [-1, 1], y(t) = y(t -τ ) = 0 y(t) max{|y(t)|,|y(t-τ )|} , otherwise .
In this case, instead of (1), the solutions of the following functional differential inclusion will be considered:

dx(t)/dt ∈ F (x t , d(t)), t ≥ 0, (2)
and for any initial condition x 0 ∈ C [-τ,0] the set of corresponding solutions of (2) initiated at x 0 for the input d can be denoted as S(x 0 , d).

Remark 1. Considering solutions of the system (1) for

f (φ, d) ≡ f 0 (φ(0), φ(-τ ), d) on the interval [0, τ ] we can derive the Filippov differential inclusion for f 0 [t, x(t), d(t)] = f 0 [x(t), x 0 (t -τ ), d(t)
] in the right-hand side. The obtained inclusion satisfies all conditions of existence theorem [START_REF] Filipov | Differential Equations with Discontinuous Righthand Sides[END_REF] implying existence of a solution locally. Having the solution defined on [0, τ ] we can repeat the same considerations for t ∈ [τ, 2τ ], etc. This method of steps allows solutions of (2) to be defined for t > 0.

For a locally Lipschitz continuous function V : R n → R + (where R + = {s ∈ R : s ≥ 0}) let us introduce the upper directional Dini derivative along the trajectories of (2):

D + F (xt,d) V [x t (0)] = sup v∈F (xt,d) limsup h→0 + V [x t (0) + hv] -V [x t (0)] h .

B. Stability definitions

Let Ω be an open neighborhood of the origin in C [-τ,0] , and denote D set of all essentially bounded measurable input d : R + → R m with ||d|| ∞ < D for some 0 < D < +∞. Combining the results given in [START_REF] Moulay | Finite-time stability and stabilization of time-delay systems[END_REF], [START_REF] Pepe | On saturation, discontinuities, and delays, in iiss and iss feedback control redesign[END_REF], [START_REF] Kolmanovskii | Introduction to the theory and applications of functional differential equations[END_REF] we obtain the following stability notions.

Definition 1. The (trivial solution x(t) = 0 of) system (2) for d = 0 is said to be (a) stable if there is σ ∈ K such that for any x 0 ∈ Ω all solutions in S(x 0 , 0) are defined for all t ≥ 0 and for all x(t, x 0 , 0) ∈ S(x 0 , 0), |x(t, x 0 , 0)| ≤ σ(||x 0 ||) for all t ≥ 0; (b) asymptotically stable if it is stable and lim t→+∞ |x(t, x 0 , 0)| = 0 for any x 0 ∈ Ω and all x(t, x 0 , 0) ∈ S(x 0 , 0); (c) finite-time stable if it is stable and for any x 0 ∈ Ω there exists 0 ≤ T x0 < +∞ such that x(t, x 0 , 0) = 0 for all t ≥ T x0 and all x(t, x 0 , 0) ∈ S(x 0 , 0). The functional T 0 (x 0 ) = inf{T x0 ≥ 0 : x(t, x 0 , 0) = 0 ∀t ≥ T x0 } is called the settling time of the system (1).

If Ω = C [-τ,0] , then the corresponding properties are called global stability/asymptotic stability/finite-time stability.

If the above properties hold for all x(t, x 0 , d) ∈ S(x 0 , d) with any x 0 ∈ Ω and any d ∈ D, then the corresponding properties are called uniform stability/asymptotic stability/finite-time stability.

For the forthcoming analysis we will need Lyapunov-Razumikhin theorem, which is given below (the variants for ordinary differential equations are given in [START_REF] Kolmanovsky | Stability of functional differential equations[END_REF], [START_REF] Gu | Stability of Time-Delay Systems[END_REF]).

Theorem 1. Let α 1 , α 2 ∈ K and η : R + → R + be a continuous nondecreasing function. If there exists a Lipschitz continuous function V : R n → R such that

α 1 (|x|) ≤ V (x) ≤ α 2 (|x|), ∀x ∈ R n ,
and the derivative of V along a solution x(t) of (2) satisfies

max θ∈[-τ,0] V [x(t + θ)] ≤ V [x(t)] ⇒ D + F (xt,0) V [x(t)] ≤ -η(|x(t)|), (3)
then (2) is stable. If, in addition, η ∈ K and there exists a continuous nondecreasing function p(s) > s for s > 0 such that the condition (3) is strengthened to

max θ∈[-τ,0] V [x(t + θ)] ≤ p{V [x(t)]} ⇒ D + F (xt,0) V [x(t)] ≤ -η(|x(t)|), then (2) is asymptotically stable. If in addition α 1 ∈ K ∞ , then (2
) is globally asymptotically stable. If the above properties are satisfied not only for d = 0 but for any d ∈ D, then (2) is uniformly stable/asymptotically stable.

Since the conditions of Theorem 1 are formulated for a Lyapunov function V , then the proof is the same for a differential equation or inclusion (it is omitted in this note).

Similarly, in order to analyze robustness of functional differential inclusions we will use an extension of the input-to-state stability (ISS) framework proposed in [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF], [START_REF] Pepe | On saturation, discontinuities, and delays, in iiss and iss feedback control redesign[END_REF].

Definition 2. The system (2) is called ISS, if there exist β ∈ KL and γ ∈ K such that for all t ≥ 0 it holds

|x(t, x 0 , d)| ≤ β(||x 0 ||, t) + γ(||d|| ∞ )
for all x 0 ∈ C [-τ,0] , all essentially bounded measurable inputs d and all x(t, x 0 , d) ∈ S(x 0 , d). Definition 3. A locally Lipschitz function V : R n → R + is called ISS Lyapunov-Razumikhin function for (2), if there exist α 1 , α 2 ∈ K ∞ and χ τ , χ d , α ∈ K such that the following conditions hold:

α 1 (|x|) ≤ V (x) ≤ α 2 (|x|) ∀x ∈ R n ; V (x(t)) ≥ χ τ [ sup θ∈[-τ,0] V {x(t + θ)}] + χ d (|d(t)|) ⇒ D + F (xt,d) V [x(t)] ≤ -α(V [x(t)]),
for all solutions of (2) x(t, x 0 , d) ∈ S(x 0 , d) for all x 0 ∈ C [-τ,0] and all essentially bounded measurable inputs d.

Theorem 2. [START_REF] Teel | Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem[END_REF] If there exists an ISS Lyapunov-Razumikhin function V for (2) and χ τ (s) < s for all s > 0, then the functional differential inclusion (2) is ISS from d to x with the gain γ(s

) = α -1 1 • χ d (s).

III. APPROXIMATION OF THE FIRST ORDER SLIDING MODE ALGORITHM

A. The rate of convergence

First of all, let us consider the system ẋ(t) = -ksign τ (x t )

for x 0 ∈ C [-τ,0] with τ > 0 and k > 0, in order to evaluate the rate of convergence in a system with the introduced approximation of the sign function, in this case F (x t , 0) = -ksign τ (x t ). First of all note that for (4) the origin (0 ∈ C [-τ,0] with 0(s) = 0 for all s ∈ [-τ, 0]) is the equilibrium: once x(t ) = 0 for some t ≥ 0, then x(t) = 0 for all t ≥ t . For any 4) it has the derivative Moreover, V (t) = D + F (xt,0) V [x(t)] < 0 wherein x(t) = 0, thus V (t) and |x(t)| are strictly decreasing. Therefore, sign[x(t)] = sign[x(t -τ )] and |x(t)| < |x(t -τ )|, x(t -τ ) = 0 for all t ≥ τ , then

x 0 ∈ C [-τ,0] consider a Lyapunov function candidate V [x(t)] = 1 2 x 2 (t), for (
D + F (xt,0) V [x(t)] = -k x 2 (t) max{|x(t)|, |x(t -τ )|} = -kα(t)|x(t)| ≤ 0, α(t) =    1, |x(t)| ≥ |x(t -τ )| |x(t)| |x(t-τ )| , |x(t)| < |x(t -τ )| . Let max θ∈[-τ,0] V [x(t + θ)] ≤ pV [x(t)] for some p > 1, then by definition of V [x(t)] we obtain |x(t -τ )| ≤ √ p|x(t)|,
x(t) = x(t -τ ) - ˆt t-τ k x(s) max{|x(s)|, |x(s -τ )|} ds
|x(t)| = |x(t -τ )| - ˆt t-τ k |x(s)| |x(s -τ )| ds ≤ |x(t -τ )| - |x(t)| |x(t -2τ )| ˆt t-τ kds = |x(t -τ )| -kτ |x(t)| |x(t -2τ )| .
Thus for t ≥ 2τ we obtain

|x(t)| ≤ |x(t -τ )||x(t -2τ )| |x(t -2τ )| + kτ
and consider the sequence of

a i = |x(iτ )| for i ≥ 1, then a i+1 ≤ a i a i-1 a i-1 + kτ , i ≥ 1. (5)
The rate of convergence of this sequence is polynomial, and for small amplitudes of x(t), the state of (4) converges to zero with a hyper-exponential speed. To confirm this conclusion, the results of this sequence calculation for different initial values of a 0 = a 1 and for kτ = 1 are shown in Fig. 1 in logarithmic scale.

Thus the following result has been proven.

Lemma 1. The system (4) is globally asymptotically stable, and it has a hyper-exponential rate of convergence.

Consequently, the proven time of convergence in (4) is not finite, and by this the convergence of ( 4) is slower than in the original system

ẋ(t) = -ksign[x(t)], (6)
but anyway it is much faster than an exponential rate, which can be obtained using standard approximations (rather restricted possibilities of finite-time convergence for time-delay systems are discussed in [START_REF] Efimov | Comments on finite-time stability of time-delay systems[END_REF]).

Another consequence of this result is that in (4) there is no chattering of the right-hand side (the right hand-side of (4) is discontinuous for x(t) = x(t -τ ) = 0, but that state is reached asymptotically). In other words, for all solutions of (4) with nonzero initial conditions, i.e. x(s) = 0 for all s ∈ [-τ, 0], the value of provided approximation for control law sign τ (x t ) is continuous for t ∈ [0, +∞) (this fact can also be seen in Fig. 2).

Of course, appearance of chattering is also dependent on the method of calculation of solutions and discretization step. Advantage of ( 4) is that it admits a simple discretization by explicit 4) and [START_REF] Levant | Chattering analysis[END_REF] or implicit (see also [START_REF] Acary | Implicit Euler numerical simulations of sliding mode systems[END_REF])

x(t i+1 ) = x(t i ) - khx(t i ) max{|x(t i )|, |x(t i -τ )|}
x(t i ) = 1 + kh max{|x(t i+1 )|, |x(t i -τ )|} x(t i+1 )
Euler method, where t i = hi, i ≥ 0 are the discrete times and h is the discretization step. Since expression in the brackets is always bigger than 1, for t i ≥ τ the implicit Euler discretization becomes explicit:

x(t i+1 ) = 1 + kh |x(t i -τ )| -1 x(t i ),
which is clearly converging for any h > 0 (that is not true for the explicit Euler discretization). For an illustration, the results of simulation of ( 4) and ( 6) with h = 3 × 10 -5 are given in Fig. 2 for x(s) = 1, s ∈ [-τ, 0] and k = 1 and τ = 0.15. The red lines represent the variables |x(t)| and ẋ(t) generated by the system (4) (explicit and implicit discretizations give the same results), the blue lines correspond to the variables of the system (6).

B. Uniform and robust stability/convergence

Now let us consider the perturbed version of (4):

ẋ(t) = -ksign τ (x t + d 1t ) + d 2 (t), (7)
where d 1 (t) ∈ R and d 2 (t) ∈ R represent the measurement noise and the additive disturbance, respectively (they are locally bounded measurable functions of time). As usual in the sliding mode theory, we will be interested in the case when ||d 2 || ∞ < D and k > D, in this case

ẋ(t) ∈ F (x t , d 1t ), (8)
F (x t , d t ) = -ksign τ (x t + d t ) + [-D, D]. Lemma 2. Let √ 2b-1 √ 2bp -1 +1 k > D for some 0 < p < 1 and b > 0.5. Then (8) is ISS from d 1 (t) to x(t) with the gain γ(s) = 2 √ 2bs.
Proof. Consider for (8) a Lyapunov function candidate V [x(t)] = 1 2 x 2 (t):

D + F (xt,d1t) V [x(t)] ≤ -kx(t)sign τ (x t + d 1t ) + |x(t)|D.
Under conditions of the lemma, there exists an > 0 such that

√ 2b-1 √ 2bp -1 +1 k ≥ D + , where 0 < p < 1 and b > 0.5,
and in order to apply Theorem 2 assume that the following restriction is satisfied on trajectories of (8):

V [x(t)] ≥ b|d 1 (t)| 2 + b|d 1 (t -τ )| 2 + p 2 V [x(t -τ )] (9)
≥ max{b|d 1 (t)| 2 , b|d 1 (t -τ )| 2 , p 2 V [x(t -τ )]}, then |x(t)| ≥ max{ √ 2b|d 1 (t)|, √ 2b|d 1 (t -τ )|, p|x(t -τ )|]}
and it is necessary to prove that D + F (xt,d1t) V [x(t)] < 0 under [START_REF] Ambrosino | Robust model tracking control for a class of nonlinear plant[END_REF]. In this case sign[x(t) + d 1 (t)] = sign[x(t)] and

D + F (xt,d1t) V [x(t)] ≤ -k|x(t)|g(x t , d 1t ) + |x(t)|D,
where

g(xt, d1t) = |x(t) + d1(t)| max{|x(t) + d1(t)|, |x(t -τ ) + d1(t -τ )|} . Obviously, g(x t , d 1t ) = 1 if |x(t) + d 1 (t)| ≥ |x(t -τ ) + d 1 (t -τ )|. Consider the case |x(t) + d 1 (t)| < |x(t -τ ) + d 1 (t -τ )|, then g(x t , d 1t ) = |x(t) + d 1 (t)| |x(t -τ ) + d 1 (t -τ )|
and under the imposed restrictions [START_REF] Ambrosino | Robust model tracking control for a class of nonlinear plant[END_REF],

|x(t)+d 1 (t)| ≥ (1-1 √ 2b )|x(t)| with |x(t-τ )+d 1 (t-τ )| ≤ [p -1 + 1 √ 2b ]|x(t)|, hence g(x t , d 1t ) ≥ √ 2b-1 √ 2bp -1 +1 in this case. Therefore, since 0 < √ 2b-1 √ 2bp -1 +1 ≤ p < 1
for any 0 < p < 1 and b > 0.5, the following relation has been proven under (9):

D + F (xt,d1t) V [x(t)] ≤ D -k √ 2b -1 √ 2bp -1 + 1 |x(t)| ≤ -|x(t)| = -2V [x(t)].
Thus, V is an ISS Lyapunov-Razumikhin function for [START_REF] Edwards | Continuous higher order sliding mode control based on adaptive disturbance compensation[END_REF] with respect to an extended input [d 1 (t) d 1 (t -τ )] T for α 1 (s) = α 2 (s) = 0.5s 2 , α(s) = √ 2s, χ τ (s) = p 2 s and χ d (s) = bs 2 . Since χ τ (s) < s for all s > 0, then according to Theorem 2 the functional differential inclusion (8

) is ISS from [d 1 (t) d 1 (t -τ )] T to x(t) with the gain γ(s) = α -1 1 • χ d (s).
Taking in mind definition of ISS property, it implies that (8) is ISS from d 1 (t) to x(t) with the gain γ(s

) = α -1 1 • χ d (2s).
For [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] the result of Lemma 2 implies that for proposed k > D the system is ISS with respect to measurement noise d 1 uniformly on matched disturbances ||d 2 || ∞ < D. Therefore, if d 1 = 0 (there is no noise), then any disturbance ||d 2 || ∞ < D will be rejected in [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]. The parameter b defines the gain γ of the system (7) with respect to measurement noise d 1 and, to this end, it determines the value of k.

The results of simulation with the discretization step h = 3×10 -5 in the Euler method for [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] and the conventional sliding mode system

ẋ(t) = -ksign[x(t) + d 1 (t)] + d 2 (t) (10)
are shown in figures 3 and 4 for τ = 0.15 and k = 1 (in this case b = 10 3 , p = 0.96 and = 10 -2 for the given D = 0.9). In the top of Fig. 3 1.5 second of time). As we can conclude, the system (7) has less chattering. In Fig. 4 the results of simulation are shown for d 1 (t) = 0.1 sin(11t) and the same d 2 (t), the variables x(t) and u(t) are shown for (7) (red lines) and (10) (blue lines). In this case, again, the variable x(t) has a similar behavior for ( 7) and [START_REF] Burton | Continuous approximation of variable structure control[END_REF], but u(t) contains less chattering for [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF].

IV. APPROXIMATION OF HIGH ORDER SLIDING MODE ALGORITHMS

The main idea of the proof of Lemma 2 has the following interpretation: if for sign[x(t) + d 1 (t)] the sliding mode system has an ISS Lyapunov function V , and since sign τ (x t + d 1t ) can be reduced to sign[x(t) + d 1 (t)] under the Lyapunov-Razumikhin arguments, then V is an ISS Lyapunov-Razumikhin function for the system with sign τ (x t + d 1t ). Thus a simple extension for high order sliding mode systems can be obtained.

Consider a conventional sliding mode system:

ẋ(t) ∈ G x(t), sign{s[x(t)]}, d(t) , (11)
where x(t) ∈ R n is the state, d(t) ∈ R m is the disturbance, G : R n+m+k → R n and s : R n → R k are continuous functions, s(t) = s[x(t)] ∈ R k is the sliding surface variable (application of the sign function to a vector argument is understood elementwise).

Assumption 1. Let the system (11) be ISS from d to x and there exist an ISS Lyapunov function

V : R n → R + such that ∀x ∈ R n and ∀d ∈ R m α 1 (|x|) ≤ V (x) ≤ α 2 (|x|); V (x(t)) ≥ χ d (|d(t)|) ⇒ D + G(x(t),sign{s[x(t)]},d) V [x(t)] ≤ -α(V [x(t)])
for some α 1 , α 2 ∈ K ∞ and χ d , α ∈ K.

Let us introduce another time-delayed modification of the sign function for the high order case:

sign τ (x t ) =                      s[x(t)] max{|s[x(t)]|,|s[x(t-τ )]|} , V [x(t)] < pV [x(t -τ )] and |s[x(t)]| = 0 s[x(t)] |s[x(t)]| , V [x(t)] ≥ pV [x(t -τ )] and |s[x(t)]| = 0 [-1, 1] k , |s[x(t)]| = 0
for some constant 0 < p < 1, where V has been introduced in Assumption 1. As before, sign 0 (x t ) = sign[x(t)].

Theorem 3. Let Assumption 1 be satisfied for the system [START_REF] Canudas-De Wit | Smoothing strategies for high-gain control[END_REF]. Then for any 0 < τ < +∞ the system

ẋ(t) ∈ G x(t), sign τ (x t ), d(t) , (12)
is also ISS from d to x.

Proof. In order to apply Theorem 2, let for 0 < p < 1 and V (x) from Assumption 1 the following relation be satisfied for the system (12):

V (x(t)) ≥ pV (x(t -τ )) + χ d (|d(t)|), then V (x(t)) ≥ max{pV (x(t -τ )), χ d (|d(t)|)}
and according to the definition of sign τ (x t ) we have sign τ (x t ) = sign{s[x(t)]} and

D + G(x(t),sign τ (xt),d) V [x(t)] ≤ -α(V [x(t)]).
Thus V is an ISS Lyapunov-Razumikhin function for [START_REF] Khalil | Nonlinear Systems[END_REF] for χ τ (s) = ps. Since χ τ (s) < s for all s > 0, then according to Theorem 2 the functional differential inclusion ( 12) is ISS from d to x with the gain γ(s

) = α -1 1 • χ d (s).
For an illustration of the result of this theorem consider the super-twisting algorithm:

ẋ1 (t) = -k 1 |x 1 (t)|sign[x 1 (t)] + x 2 (t), (13)
ẋ2 (t) = -k 2 sign[x 1 (t)] + d(t), where 0 < k 1 < k 2 are parameters, G (x, σ, d) = -k 1 |x 1 |sign[x 1 ] + x 2 -k 2 σ + d .
As a Lyapunov function candidate take one from [START_REF] Polyakov | Reaching time estimation for "super-twisting" second order sliding mode controller via lyapunov function designing[END_REF] (the Lyapunov function proposed in [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF] also can be used) 5. The red solid lines represent the variables of time-delayed algorithm, the blue dash lines correspond to the variables of ( 13), the variable u(t) in both cases demonstrate the function sign(•) behavior and its approximation. As we can conclude from this numerical experiment, the higher order sliding mode algorithm behavior is quite nice and smooth (see the variables x 1 (t) and x 2 (t)), but even in this case a chattering reduction is achieved by a slight decrease of the convergence rate using the new continuous delayed sliding mode algorithm.

V. CONCLUSIONS

In this note the Delayed Sliding Mode control has been presented, which demonstrates how by a delay introduction in the closed-loop of a nonlinear system it is possible to improve the quality of steady-state behavior. An approximation of the sign function has been given based on delayed values of the processed variable, which ensures chattering reduction for the sliding mode control and estimation algorithms. The delay has been introduced in a way that under the Lyapunov-Razumikhin conditions the system is reduced to its original non-delayed form, then its stability can be easily concluded. Efficiency of the proposed approach has been demonstrated on examples: the chattering can be avoided (see Fig. 2) or tolerated (see figures 3-5) by a slight decreasing of the convergence rate, in addition, the matched disturbances are compensated. Influence of discrete-time realization on the obtained in continuous time properties, tuning rules and asymptotic performance analysis are directions of future research.

  or equivalently, α(t) ∈ [p -0.5 , 1], and D + F (xt,0) V [x(t)] ≤ -kp -0.5 |x(t)| that by Theorem 1 implies global asymptotically stability for (4).

Figure 1 .

 1 Figure 1. The results of the sequence (5) calculation

Figure 2 .

 2 Figure 2. Results of simulation for (4) and[START_REF] Levant | Chattering analysis[END_REF]

Figure 3 .xFigure 4 .

 34 Figure 3. Trajectories of (7) (red) and (10) (blue) for d1 = 0

1 Figure 5 .

 15 Figure 5. Results of simulation for super-twisting algorithm

  the variable x(t) is shown for both[START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] (red line) and (10) (blue line), for d 1 (t) = 0 and d 2 (t) = 0.3[sin(15t) + sin(πt) + 1]. At the bottom of the figure, on the left the variable u(t) = sign τ (x t + d 1t ) for (7) (red line) and the signal -d 2 (t) (green dash line) are plotted (the variable u(t) represents a kind of "control" approximation in a sliding mode control system), on the right the variable u(t) = sign τ (x

	x(t)
		t
	u(t)
	d 2 (t)	u(t)
	t	t

t + d 1t ) for

[START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF]

and the variable u(t) = sign[x(t) + d 1 (t)] for

[START_REF] Burton | Continuous approximation of variable structure control[END_REF]

are shown (the latter one presents a pure chattering after approximately

This work was supported in part by ANR Grant CHASLIM (ANR-11-BS03-0007); by CNRS/CONACYT projects 222629, 209731; by CONACyT 261737; by the Government of Russian Federation (Grant 074-U01) and the Ministry of Education and Science of Russian Federation (Project 14.Z50.31.0031); by Programa de Apoyo a Proyectos de Investigacion e Innovacion Tecnologica (UNAM) 113613, DGAPA PASPA Program. is much faster than any exponential [1].