Resveratrol

(3,4′,5-trihydroxy-transstilbene

essential oils are the plant extracts that have been the most studied for their antimicrobial activity. However, in addition to their higher cost than other plant extracts (due to their lower extractability and yield), essential oils contain volatile compounds (frequently unpleasant or generating off-odors) and have low water solubility. These drawbacks often limit their use for food preservation. Since applications of essential oils for improving food safety have been widely reviewed by Burt 2 and more recently by Hyldgaard et al., [START_REF] Hyldgaard | Essential oils in food preservation: mode of action, synergies, and interactions with food matrix components[END_REF] essential oils are outside the scope of this review, which will instead be focused on other plant extracts.

Plant extracts play an important role because of their nutritional, visual (color), and taste properties; polyphenols are thus considered relevant due to their qualities. [START_REF] Robins | Phytochemistry of Fruit and Vegetables[END_REF] Most of the applications of phenolic-rich plant extracts for food safety purposes are related to their antioxidant activity, including the prominent example of rosemary extract, [START_REF] Cuvelier | Antioxidative activity and phenolic composition of pilot-plant and commercial extracts of sage and rosemary[END_REF] which has food additive status in the European Union (E 392). Antioxidant rosemary extract contains more than 900 g kg -1 of carnosic acid and carnosol, both of which are phenolics.

In addition to these antioxidant properties, polyphenols have also been shown to extend the shelf life of some food products through their antimicrobial activities, and they may also act as inhibitors of pathogenic microorganisms. [START_REF] Ba | Selecting new peach and plum genotypes rich in phenolic compounds and enhanced functional properties[END_REF] This review will thus help the reader to identify (i) phenolic-rich plant extracts with potential as antimicrobials, (ii) their mechanism of action, (iii) the factors known to affect their in situ antimicrobial activity in real food systems, and (iv) most of their potential applications for perishable food preservation reported up to now.

NATURAL EXTRACTS/MOLECULES FOR FOOD PRESERVATION

Edible plants: a potential source of antimicrobial molecules Controlling microbial growth in food products has always been a major concern for the different stakeholders in the agri-food sector. A double challenge must be considered: ensuring both food safety and food waste reduction. In fact, microbes causing infectious diseases are frequently the cause of morbidity and mortality across the world. In addition, microbial spoilage induces the loss of approximately a quarter of the world's food supply, and more than 40% of food damage occurs at the retail and consumer levels in developed countries. [START_REF]Global food losses and food waste-extent, causes and prevention[END_REF] This context has induced a rise in biocide and antibiotic application in order to guarantee efficient control of the microbial contamination of foods.

The emergence and spread of antibiotic resistance among human pathogenic microorganisms are a critical challenge. Indeed, the appearance of resistance or even multiresistance in a large bacterial community can be induced by the routine use of antibiotics. [START_REF] Chen | Microbiological safety of chicken litter or chicken litter-based organic fertilizers: a review[END_REF] In addition to this phenomenon of antibiotic resistance, the presence of antimicrobial agent residues in the environment has attracted much attention from modern consumers. Thus, the search for natural antimicrobials that are effective against both pathogenic and spoilage microorganisms is crucial.

Natural preservatives are considered healthier and to have an added value arising from their bioactivity and nutritional value. Therefore, an increasing number of food companies have made an effort to meet the increasing consumer demand for natural food preservatives. However, a soft transition from chemical additives to natural alternatives is expected, particularly because of economic and antimicrobial efficacy issues that still have to be solved. [START_REF] Carocho | Adding molecules to food, pros and cons: a review on synthetic and natural food additives[END_REF] Plant-derived antimicrobials are promising in this context. Indeed, plant extracts are generally considered edible based on their traditional human consumption. In addition, plant secondary metabolites (PSMs) account for the greatest diversity of structures (e.g. there are more than 12 000 known alkaloids, more than 10 000 phenolic compounds and over 25 000 different terpenoids). Different antimicrobial polyphenol subgroups, their chemical structures and examples are presented in Fig. 1.

Antimicrobial PSMs were thus proposed as potential alternatives to synthetic preservatives. Nevertheless, plant-derived antimicrobials have not been frequently applied until now. To expand the use of plant extracts as natural preservatives, their bioactive compounds could be extracted and purified by developing economic processes preserving their activities.

Typically, extraction of phenolic compounds as a mixture and their purification are simple to perform. The edible plant-derived compounds remain the most favored for food use to limit concern regarding toxicity. Therefore, several recent studies have explored these plants as a potential source of antimicrobial molecules. The in vitro antimicrobial activity of the most studied edible plants byproducts in the past decade is described in Table 1.

Polyphenols: a diversity of antimicrobial mechanisms of action

The OH groups of phenolic compounds interact with the cell membrane of bacteria by hydrogen bonding. Importantly, the presence of OH functional groups is relevant to the antibacterial activity of many phenolics. [START_REF] Lai | Antimicrobial and chemopreventive properties of herbs and spices[END_REF][START_REF] Knobloch | Progress in Essential Oil Research[END_REF][START_REF] Dorman | Antimicrobial agents from plants: antibacterial activity of plant volatile oils[END_REF][START_REF] Ultee | The phenolic hydroxyl group of carvacrol is essential for action against thefood-borne pathogen Bacillus cereus[END_REF] Indeed, the interaction of phytochemicals with bacterial cell membranes usually causes either the disruption of the membrane structure, which induces loss of cellular content, [START_REF] Knobloch | Progress in Essential Oil Research[END_REF][START_REF] Sikkema | Interactions of cyclic hydrocarbons with biological membranes[END_REF] or the delocalization of electrons (because of the double bonds of the aromatic nucleus), which results in depolarization of bacteria (acting as proton exchangers) and thus affects the proton motive force, reducing the pH gradient across the membrane and the level of the ATP pool. [START_REF] Ultee | The phenolic hydroxyl group of carvacrol is essential for action against thefood-borne pathogen Bacillus cereus[END_REF] This series of mechanisms initiated by the active hydroxyl groups of phytochemicals such as phenolics can lead to cell death. Furthermore, the relative position of the OH group on the phenolic nucleus has been reported to influence the antibacterial efficacy of phenolic compounds. [START_REF] Gyawali | Natural products as antimicrobial agents[END_REF] The presence of alkyl groups in the aromatic nucleus generates phenoxyl radicals [START_REF] Pauli | Inhibitory effects of essential oil components on growth of food-contaminating fungi[END_REF] reported to enhance the antibacterial efficacy of phenolics and alter their distribution ratio between aqueous and non-aqueous phases, including bacterial phases. [START_REF] Pelczar | Microbiology[END_REF] The presence of an acetate moiety in the molecular structure appeared to increase the activity of parent phenolic compounds by either the alcohol groups as protein denaturing agents [START_REF] Pelczar | Microbiology[END_REF] or by an increase in their electronegativity due to the aldehyde groups promoting electron transfer and reactions with membrane proteins. [START_REF] Kurita | Antifungal activity and molecular orbital energies of aldehyde compounds from oils of higher plants[END_REF] The galloyl moiety has been reported to induce important damage to the membrane structure, thereby enhancing the antibacterial activity of epigallocatechin gallate against Gram-positive bacteria. [START_REF] Ikigai | Bactericidal catechins damage the lipid bilayer[END_REF] In addition to their chemical composition and structure, the lipophilic properties of phytochemicals are also involved in their antibacterial activity. [START_REF] Pauli | Inhibitory effects of essential oil components on growth of food-contaminating fungi[END_REF] The antimicrobial activity of many phenolics has been observed to increase with the elevation of their lipophilic character; this may be directly related to their potential interactions with the cell membrane. [START_REF] Ikigai | Bactericidal catechins damage the lipid bilayer[END_REF] The ability to penetrate the cell membrane and interact with cell compounds induces irreversible damage to the cell membrane and coagulation of the cell content, affecting both membrane and intracellular enzymes. Hydrogen-bonding descriptors (H-bond donors, H-bond acceptors), polar surface area, log P (octanol/water partition coefficient) and HOMO (highest occupied molecular orbital) and LUMO (lowest unoccupied molecular orbital) energy levels are the physicochemical parameters involved in the capacity of bioactive molecules to permeate through lipid membranes. [START_REF] Maherani | Calcein release behavior from liposomal bilayer; influence of physicochemical/mechanical/structural properties of lipids[END_REF][START_REF] Fichert | A structure-permeability study of small drug-like molecules[END_REF] Van der Waals bonding can be established when the electron bonding energy of phenolics (acting as hydrogen bond donors) is higher than that of membrane lipids. A high number of hydrogen bond donors leads to higher interactions with the membrane of bacteria. Furthermore, molecules with a polar surface area of greater than 140 Å squared (higher than acyl chains) tend to interact strongly with the choline head groups of membrane phospholipids. [START_REF] El Maghraby | Drug interaction and location in liposomes: correlation with polar surface areas[END_REF] Strong interactions with the Caco-2 cell model membrane have been observed for molecules with log P values higher than 0 and less than 3, 68 allowing them to get closer to membrane surfaces and interact with them; however, such data regarding bacterial membranes are lacking. Attraction between two different molecules can be induced by the interaction between their HOMO and LUMO. The difference in the HOMO and LUMO energy levels between phenolics and lipids constitutes a key element for bacterial membrane and phenol interactions.

The main antimicrobial mechanisms of action of polyphenols cited in this review are summarized in Fig. 2.

Not all information on the mechanisms of the antimicrobial action of phenolics has been acquired yet. However, these components are considered to have many sites of action at the cellular level. [START_REF] Radulovic | Antimicrobial plant metabolites: structure diversity and mechanism of action[END_REF] There are at least three mechanisms on which several authors agree: (i) modification of the permeability of cell membranes, formation of cytoplasmic granules and rupture of the cytoplasmic membrane; (ii) changes in various intracellular functions induced by hydrogen bonding of the phenolic compounds to enzymes through their OH groups; and (iii) modification of fungal morphology (cell wall rigidity and integrity losses) induced by different interactions with cell membranes. [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] In general, Gram-negative bacteria are more resistant to PSMs, including phenolics, than Gram-positive bacteria. [START_REF] Burt | Essential oils: their antibacterial properties and potential application in foods: a review[END_REF][START_REF] Negi | Plant extracts for the control of bacterial growth: efficacy, stability, and safety issues for food application[END_REF] This difference is likely because Gram-negative bacteria possess a cell wall linked to an outer complex membrane, [START_REF] Ratledge | Microbial Lipids[END_REF] namely, the lipopolysaccharide envelope, which slows down the passage of phytochemicals. [START_REF] Vaara | Agents that increase the permeability of the outer membrane[END_REF][START_REF] Inouye | Screening of the antibacterial effects of variety of essential oils on respiratory tract pathogens, using a modified dilution assay method[END_REF] Nevertheless, the appearance of lipopolysaccharides released from the outer membrane provides evidence that some polyphenols may also affect the outer membrane of Gram-negative bacteria. [START_REF] Helander | Characterization of the action of selected essential oil components on Gram-negative bacteria[END_REF][START_REF] Lambert | A study of the minimum inhibitory concentration and mode of action of oregano essential oil, thymol and carvacrol[END_REF][START_REF] Storia | Atomic force microscopy analysis shows surface structure changes in carvacrol-treated bacterial cells[END_REF] Additional impacts were reported when Gram-positive bacteria and fungi were considered. The first one is the modification of intracellular pH (due to variations in the flow of ions such as H + and K + influencing the proton motive force), and the second one is the blocking of energy production (interference with the energy (ATP)-generating system). [START_REF] Djilani | The therapeutic benefits of essential oils, in Nutrition, Well-being and Health in Technology[END_REF] Condensed phenylpropanoids -tannins -may induce damage at the cell membrane level and even inactivate metabolism by binding to enzymes, [START_REF] Ya | Carbohydrate-polyphenol complexation[END_REF][START_REF] Chung | Mechanism of inhibition of tannic acid and related compounds on the growth of intestinal bacteria[END_REF] while phenolic acids have been shown to disrupt membrane integrity, as they cause consequent leakage of essential intracellular constituents. [START_REF] Borges | Antibacterial activity and mode of action of ferulic and gallic acids against pathogenic bacteria[END_REF] Flavonoids are considered able to promote complex formation by linking with soluble proteins located outside the cells and within the cell walls of bacteria. [START_REF] Tsuchiya | Comparative study on the antibacterial activity of phytochemical flavanones against methicillin-resistant Staphylococcus aureus[END_REF][START_REF] Cushnie | Antimicrobial activity of flavonoids[END_REF] Quercetin was reported to have a significant effect on the bacterial membrane by increasing the membrane permeability and disturbing its potential. [START_REF] Mirzoeva | Antimicrobial action of propolis and some of its components: the effects on growth, membrane potential and motility of bacteria[END_REF] Investigations on the membrane action of several flavonoids ((-)-epigallocatechin gallate, [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] (-)-epicate chin gallate and 3-O-octanoyl-(+)-catechin, [START_REF] Stapleton | Anti-Staphylococcus aureus activity and oxacillin resistance modulating capacity of 3-O-acylcatechins[END_REF] as well as 2,4,2 ′ -trihidroxy-5 ′ -methylchalcone [START_REF] Sato | Synergistic effects of mupirocin and an isoflavanone isolated from Erythrina variegata on growth and recovery of methicillin resistant Staphylococcus aureus[END_REF] ) showed that these compounds induce a reduction in membrane fluidity.

Furthermore, some flavonoids may act by inhibiting both energy metabolism and DNA synthesis, as observed by Haraguchi et al., [START_REF] Haraguchi | Mode of antibacterial action of retrochalcones from Glycyrrhiza inflata[END_REF] and affect protein and RNA syntheses to a lesser extent. In addition to their versatile activities, catechins exert antibacterial effects via DNA gyrase inhibition. [START_REF] Gradisar | Green tea catechins inhibit bacterial DNA gyrase by interaction with its ATP binding site[END_REF] Other flavonoids, such as apigenin, have been proven to have an inhibitory effect on the activity of DNA gyrase and hydroxyacyl-acyl carrier protein dehydratase. [START_REF] Cushnie | Antimicrobial activity of flavonoids[END_REF][START_REF] Zhang | Three flavonoids targeting the hydroxyacyl-acyl carrier protein dehydratase from Helicobacter pylori: crystal structure characterization with enzymatic inhibition assay[END_REF] Naphthoquinones (e.g. plumbagin) were found to inhibit potential efflux pumps. Interestingly, they have been shown to have a significant antibacterial effect against Gram-negative bacteria, whose resistance to most natural antimicrobial products is related to efflux pumps, [START_REF] Kuete | Antibacterial activity against bacteria expressing a multidrug-resistant phenotype[END_REF] while coumarins have been reported to cause a reduction in cell respiration. [START_REF] Chung | Tannins and human health: a review[END_REF] Recent advances in multiparameter flow cytometry offer the opportunity to obtain high-speed information on the mechanism of action of antimicrobials at real time at the single-cell level. [START_REF] Léonard | Recent advances on multi-parameter flow cytometry to characterize antimicrobial treatments[END_REF] Flow cytometric analysis revealed that resveratrol-trans-dihydrodimer disturbs the membrane potential and hinders DNA synthesis of microorganisms. [START_REF] Mora-Pale | Antimicrobial mechanism of resveratrol-trans dihydrodimer produced from peroxidase-catalyzed oxidation of resveratrol[END_REF] Paulo et al. [START_REF] Paulo | Antimicrobial activity and effects of resveratrol on human pathogenic bacteria[END_REF] used microscopic analysis combined with flow cytometry to determine the bacteriostatic effect of 200 mg L -1 resveratrol (4 × minimal inhibitory concentration (MIC) for Bacillus cereus and 2 × MIC for Staphylococcus aureus at 1-2 × 10 8 CFU mL -1 ) against Gram-positive bacteria. Modifications in cell morphology and DNA were observed in the presence of resveratrol; it can thus be assumed that it interferes with the cell cycle of bacteria. Duggirala et al. [START_REF] Duggirala | Phytochemicals as inhibitors of bacterial cell division protein FtsZ: coumarins are promising candidates[END_REF] screened 11 natural phenolics for the inhibition of the bacterial division protein FtsZ and identified coumarins as promising candidates. Recently, Liu et al. [START_REF] Liu | Assessment of the antimicrobial activity of olive leaf extract against foodborne bacterial pathogens[END_REF] observed that sublethal treatments of Listeria monocytogenes cells with olive leaf extract abolished flagella and thereby reduced their motility. Other antibacterial mechanisms of action of phenolic-rich plant extracts, such as inhibition of quorum sensing, which is involved in pathogenesis of many food bacteria, cannot be excluded; such an antibacterial mechanism of action was recently reported for oregano essential oil. [START_REF] Alvarez | Oregano essential oil-pectin edible films as anti-quorum sensing and food antimicrobialagents[END_REF] The modes of action of polyphenols discussed earlier are diverse, and this can be advantageous for their application as natural antimicrobial agents. However, further studies in this field are needed to obtain the rationale for their utilization as antimicrobial food additives. For instance, transcriptomic analysis of microbial cells treated with sublethal doses of plant phenolics or plant extracts should be performed to identify genes whose expression would be modified. For instance, quantitative polymerase chain reaction (qPCR) analysis of the expression of genes in Escherichia coli O157:H7 following treatment with cranberry concentrate allowed observation of the downregulation of genes coding for bacterial membrane and cell wall constituent synthesis. [START_REF] Wu | Application of cranberry concentrate (Vaccinium macrocarpon) to control Escherichia coli O157:H7 in ground beef and its antimicrobial mechanism related to downregulated slp, hdea and cfa[END_REF]

USES OF ANTIMICROBIAL PLANT POLYPHENOLS IN FOODS Antimicrobial activity of plant polyphenols in foods

The food use of plant extracts has been shown to be promising, and some of these extracts already possess a generally recognized as safe (GRAS) status. A comprehensive document regarding the use of phytochemicals as food preservatives has been developed by the International Life Sciences Institute-Europe: [START_REF] Schilter | Guidance for the safety assessment of botanicals and botanical preparations for use in food and food supplements[END_REF] it is focused on their effective identification and characterization. However, the recent increase in natural preservative use has induced changes in European legislation (EC/1334/2008) [START_REF]99 Food and Drug Administration (FDA)[END_REF] implemented in January 2011. These changes include new statements for natural extracts and treatments used for their preparation. At the same time, there is a need to highlight processes that are more environmentally friendly than the usual methods. In the Code of Federal Regulations, Title 21, the Food and Drug Administration (FDA) 99 defines safe natural additives for food use according to the following statements: (i) they are used in the minimum quantity required to produce their intended physical or technical effect and in accordance with all the good manufacturing practice principles; (ii) in the appropriate forms (plant parts, fluid and solid extracts, concentrates, absolutes, oils, gums, balsams, resins, oleoresins, waxes, and distillates), they can be used alone or in combination with flavoring substances and GRAS adjuvants in food. Many plant-derived products have thus been proposed as food ingredients or supplements, and they take an interesting place in the market among other healthy products.

Beverages (water-and tea-based drinks, yogurts and smoothies) are the most common foods fortified with polyphenols. 100 In addition to their antioxidant activity, a great deal of effort has recently been made to include polyphenols in natural functional ingredients as food antimicrobial preservatives. 101 Plants synthesize polyphenols in response to stress because of their self-defense from diseases mainly induced by microorganisms. That is why they are considered a promising source of antimicrobials with healthy features. Many natural phenolics that are widespread in nature, especially those that are extracted from edible plants and that have proven antimicrobial activities, could be used as potential food preservatives. [START_REF] Engels | Sinapic acid derivatives in defatted Oriental mustard (Brassica juncea L.) seed meal extracts using UHPLC-DAD-ESIMS and identification of compounds with antibacterial activity[END_REF][START_REF] Plumed-Ferrer | The antimicrobial effects of wood-associated polyphenols on food pathogens and spoilage organisms[END_REF][102][103][104] The antibacterial activity of three pure compounds naturally occurring in plants, caffeic acid, p-coumaric acid, and rutin, in different food products was tested by Stojkovic et al. 104 Amounts of p-coumaric and caffeic acids above 0.1 g L -1 completely inhibited Staphylococcus aureus growth in chicken soup, and after 72 h, no cell survival was observed in samples treated with greater than 0.9 g L -1 rutin and stored at either 25 or 4 ∘ C. Phenolic compounds have a relevant role in the visual appearance (pigmentation and browning), taste (astringency) and odor (aromas) of plant-derived products. 105,106 Interestingly, sensory evaluation for overall acceptance revealed that compared to those of the control samples, the sensory features of chicken soup and pork meat exposed to p-coumaric acid and caffeic acid were well appreciated. 104 The antibacterial effect of gallic acid combined with caffeic acid, rutin and quercetin against Escherichia coli was investigated in a meat model system at 4 ∘ C. This combination of phenolics at a concentration of 100 or 200 mg L -1 was bactericidal after 14 or 21 days of incubation. Such a synergistic effect makes it possible to enhance the activity of the polyphenols and reduce their effective concentration. 107 The antimicrobial potency of pinosylvin was evaluated in different food systems: [START_REF] Plumed-Ferrer | The antimicrobial effects of wood-associated polyphenols on food pathogens and spoilage organisms[END_REF] 25-200 mg kg -1 pinosylvin caused a decrease of 2-4 log of Enterobacteriaceae in fermenting sauerkraut. The antimicrobial activity of pinosylvin (140 mg kg -1 ) against Listeria monocytogenes inoculated in fresh gravlax was higher at 8 ∘ C than at 20 ∘ C. Saccharomyces cerevisiae inoculated in strawberry jam was completely eliminated by 300 mg kg -1 pinosylvin. However, 75 mg L -1 pinosylvin was sufficient to completely inhibit Staphylococcus aureus growth in culture media, while 200 mg L -1 pinosylvin had no effect in milk.

Numerous in vitro studies have been performed in microbiological culture media to assess the antimicrobial activity of plant extracts, but far fewer studies have addressed their application to food products. The lower antimicrobial efficacy of plant extracts in real foods may be the reason for this imbalance of information. [START_REF] Negi | Plant extracts for the control of bacterial growth: efficacy, stability, and safety issues for food application[END_REF] The presence of the glycosyl groups of flavonoids contained in the crude extracts is partially responsible for the reduction in their activity against a wide range of bacteria reported in several studies. [START_REF] Kuete | Antibacterial activity against bacteria expressing a multidrug-resistant phenotype[END_REF][START_REF] Léonard | Recent advances on multi-parameter flow cytometry to characterize antimicrobial treatments[END_REF][START_REF] Mora-Pale | Antimicrobial mechanism of resveratrol-trans dihydrodimer produced from peroxidase-catalyzed oxidation of resveratrol[END_REF]108,109,110 In situ evaluation of the antimicrobial activity of plant extracts is crucial for food use because of interactions of their bioactive compounds with food components, most likely reducing their effectiveness. However, in vitro screening remains a first step to identify the antimicrobial potential of plants. An antimicrobial efficacy similar to that in in vitro cultures could be achieved by adding higher amounts of plant extract to foods. [START_REF] Knobloch | Progress in Essential Oil Research[END_REF] Two-fold, ten-fold, 50-fold, and 25-100-fold higher plant extract concentrations were necessary to have the same antimicrobial effect in skimmed milk (from 0.6 g kg -1 to 10 g kg -1 for rosemary extract), 111 in pork liver sausage (from 5 to 50 mL kg -1 for rosemary extract against Listeria monocytogenes), 112 and in soft cheese (from 0.04 to 2.5 mL kg -1 for a mixture of rosemary, sage and citrus extracts against Listeria monocytogenes), 113 respectively, as in in vitro trials. [START_REF] Burt | Essential oils: their antibacterial properties and potential application in foods: a review[END_REF]114 For instance, Miceli et al. 115 observed that a ten-fold increase in the quantity of Borago officinalis (from 10 to 100 g L -1 ) and Brassica juncea (from 3.1 to 31 g L -1 ) aqueous extracts was necessary to achieve an antimicrobial effect in meat, fish, and vegetables. This variance can result from the interactions that occur in food systems between hydrophobic bioactive constituents of plant extracts and major food ingredients such as fat and proteins. Based on their hydrophilic character, other phytochemicals behave differently in food products. The dose of phenolics applied for food preservation should be set with sensory considerations and not based solely on in situ antimicrobial efficacy. To be accepted as food preservatives, phenolic-rich plant extracts should not strongly impart their typical color and flavor to foods. Ideally, the plant extract is chosen according to culinary associations already existing in consumer behaviors (e.g. rosemary for meat, 112 thyme for vegetable and chicken soups [START_REF] Ultee | The phenolic hydroxyl group of carvacrol is essential for action against thefood-borne pathogen Bacillus cereus[END_REF] ).

In food matrices rich in fat, a lipid coating that wraps the microorganisms and protects them from antimicrobials can form. [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] Uhart et al. 116 reported that spices inactivate Salmonella Typhimurium DT104 under in vitro conditions, whereas a decrease in their inhibition efficacy was observed when the spices were included in complex food matrices (e.g. ground beef ). Similarly, green and jasmine tea did not significantly reduce Listeria monocytogenes, Staphylococcus aureus or total bacterial counts in ground beef. 117 In addition, compared to culture media, many foodstuffs have a reduced water content, which may limit the transport of antimicrobials into the microbial cells. [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] Other potential causes include modifications in the solubility and charge of phenolics and variations in the cell envelope of target bacteria. The known interaction of many polyphenols with proteins might result in polyphenol-protein complexation (as reviewed by Papadopoulou and Frazier 118 ) and thus limit the action of active polyphenolic compounds against microbial cells. Food-mimicking matrices prepared by dispersing proteins and/or fat in liquid media can help estimate the minimum concentrations inhibiting or killing microorganisms (minimum inhibitory concentrations (MICs) or minimum bactericidal concentrations (MBCs) for bacteria, respectively) in food systems.

The physiological state of target microorganisms in foods is also likely an important factor affecting the in situ efficiency of many antimicrobials: most in vitro antimicrobial activity assays are performed with microorganisms in an optimal environment without any limiting substrates to allow their exponential growth, which is not the case in real foods.

Some phenolic-containing aqueous plant extracts that have exhibited a broad antimicrobial spectrum (among those that were non-toxic and had a relatively limited odor and taste) and that have already been used for direct incorporation in foods or in food packaging materials in the past 15 years are listed in Table 2.

Stability of plant polyphenols

Polyphenol stability is a crucial property for application in food systems and is a function of several factors, such as size, chemical structure, water solubility and polarity. Recently, nutrition has become a tool to promote human health, and maximal knowledge of the effects of treatment processes is essential for maintaining the functions of plant biomolecules not only as food preservatives but also as compounds of nutritional interest. [START_REF] Negi | Plant extracts for the control of bacterial growth: efficacy, stability, and safety issues for food application[END_REF] For the application of phenolic compounds for food preservation, they have to be stable until the expiration date of the product to which they were added. However, polyphenols are relatively unstable when directly applied in foods. The stability of such compounds in food systems can be attributed to a series of stabilities: physical, chemical, colloidal, and biological, which are correlated with each other. 159 Co-extrusion of a linear low density polyethylene (LLDPE)-based film blended with grape seed extract (10 g kg -1 ) in a twin-screw extruder with a barrel temperature ranging from 160 to 190 ∘ C resulted in a strong reduction in the antimicrobial activity of the extract. 160 Conversely, a polyethylene-based film blended with pomegranate peel extract (15 g kg -1 ) and produced by the same process demonstrated good antimicrobial activity in another work. 161 Polyphenol stability in solution depends on environmental factors (e.g. pH, electrolyte composition, and presence of oxidants). The instability of phenolic compounds can occur at pH 1-11, favorable for oxidative degradation, complex formation and reactions with other phenols, amino acids, proteins, and metal ions. 162,163 However, pH changes may induce new structures and colors of phenolics (the red wine pigment malvidin 3-glucoside may change from red at pH 1 to colorless at pH 4-5, to purple at pH 6-7, and to yellow at pH 7-8 163 ).

The OH groups located on the benzene ring of phenolic compounds are often involved in responses to pH variations. Ultraviolet spectral monitoring allowed Friedman and Jurgens 163 to observe that in contrast to caffeic, chlorogenic and gallic acids (with two or three OH groups attached to the benzene ring), conjugated non-phenolic aromatic acids such as trans-cinnamic acid without any OH groups are stable at high pH (pH 11).

Other structural criteria play a role in promoting the complexation of phenolics with other solutes at high pH values and thus induce their instability. For instance, more complex phenolics with multiring aromatic structures such as catechin, epigallocatechin, and rutin have ionized and resonance forms that are more resistant to degradation by pH than monocyclic compounds. 163 The number of OH groups located on the benzene ring can indicate the ability of phenolics to form quinone oxidation products. Ferulic acid with no more than one OH group is stable at high pH (pH 7-11), whereas caffeic acid is unstable to pH variations because of the two adjacent phenolic OH groups on the benzene ring. 163 Not only the number of OH groups but also their presence on the same or separate aromatic rings and their position (meta-or ortho-position) affect the ability of phenolics to interact with each other via conjugation or quinone formation. The spatial arrangement of OH groups is reported to influence the stabilities to pH variations. 163,164 The phenolic acids caffeic acid, chlorogenic acid, and gallic acid have been found to be irreversibly affected by high pH values. 163 Conversely, an acidic pH had no effect on chlorogenic acid stability after inclusion in apple juice. Other phenolics, such as (-)-catechin, (-)-epigallocatechin, ferulic acid, rutin, and trans-cinnamic acid, may resist pH-induced degradation. 163 Concerning anthocyanins, it is well known that variations in pH significantly influence their stability and color. 163 Curcumin is primarily used as a food additive (coloring agent: E100 (i)), but it is increasingly considered a multifunctional bioactive molecule. Interestingly, glycosylated 165 or amino acid-conjugated 166 curcumin have shown similar antibacterial, antioxidant and antimutagenic activities as pure curcumin, so such chemical modifications did not affect the earlier mentioned bioactive properties of curcumin. However, microcapsules of curcumin (prepared with gelatin and porous starch by the spray-drying method) have better solubility and stability than free curcumin along with similar antibacterial and antifungal activities. 167 Colloidal stability is defined as the ability of polyphenols to maintain a homogeneous dispersion in food matrices under various storage conditions. The stability of polyphenols added to complex food matrices can be predicted based on information on various interactions that can occur with surrounding components present in foods. Repulsive forces among charged groups may prevent polymerization and aggregation of these active compounds. Through the choice of the appropriate formulation of phenolic extracts, the formed electrostatic repulsive forces may thus increase the stability of the system once incorporated into the foods.

Biological stability includes the ability of polyphenols to preserve their antioxidant and antimicrobial properties after processing and for long-term storage under the same conditions. Room or refrigerated temperatures are the environmental conditions in which polyphenol activities are usually evaluated. To preserve their bioactivity, polyphenols can be freeze-dried. Many bioactive molecules can undergo chemical degradation, isomerization or polymerization during harsh food processes such as baking, steaming and extrusion, thereby possibly inducing a loss of their activities. 101 When incorporated into food matrices, phenolic compounds may be subjected to temperature variations. The matrix nature is the most influential factor in the thermal stability of botanical compounds in foods. Normally, phenolic compounds with higher melting temperatures are more stable to heat processing, but the effect of heat can be more pronounced in the presence of other food ingredients. A pure aqueous solution of chlorogenic acid (207-209 ∘ C melting temperature) has been found to be stable to heat treatment (1 h at 90 ∘ C) 163 , while the loss (leaching out or decomposition) of chlorogenic acid heated (30 min at 100 ∘ C or 18 min at 121 and 204 ∘ C) in the presence of food constituents was reported by other authors. 168,169 Heat-treated (100 ∘ C, 15 min) drumstick leaf extracts showed a significant decrease in their antioxidant activity compared to that of untreated samples. In contrast, the antioxidant activity of carrot tuber extract was not affected by the same heat treatment. 170 In some cases, heat treatment (105 ∘ C, 20 min) induced the formation of new molecules, which either reduce, preserve or even improve the antimicrobial activity of different plant extracts. 171,172 For instance, new phenolics with low molecular weights were found in heated grape seed extracts. 173 However, it was reported that the polyphenol content of foods decreases in response to thermal processing and long-term storage. 174,175 The thermal stability of polyphenols in apple juice was studied by Spanos et al. 176 and van der Sluis et al. 177 Cinnamic acid, procyanidin and quercetin contents have been found to decrease when apple juice was stored at room temperature. Compared to freeze-dried grape seeds, grape seeds that were heat-dried at 100 and 140 ∘ C exhibited 18.6% and 32.6% decreases in total polyphenols, respectively. 178 Heating at 60 ∘ C or above for 8 h dramatically reduced the procyanidin and anthocyanin contents in freeze-dried grape pomace. 179 Some studies have suggested heat-stable plant extracts that could be used as food preservatives, such as cinnamon in cookies; 180 Garcinia extract; 181 grape, amla, and drumstick leaf extracts 182 in biscuits; and mango fiber concentrate (with 16.1 mg g -1 of soluble polyphenols) in bread and cookies. 183 Successful food product development is deeply based on the nature of the interactions that may occur between the ingredients of food. 101 A strong interaction between grape seed procyanidins and proteins leads to the formation of protein-tannin aggregates. The molecular weight and polymerization degree of procyanidins increase this aggregation. 184 Carvalho et al. 185 indicated that this type of binding is influenced by the protein properties (molecular size, hydrophobicity and structural flexibility), the polyphenol properties (degree of polymerization, extent of galloylation, structural flexibility) and environmental factors (temperature, pH, ionic strength, presence of organic solvents and presence of carbohydrates). Since the astringent taste of polyphenol-rich fruits and vegetables, 106 haze in beverages 186 and the bioavailability reduction in both food protein and polyphenols 118,187 depend on tannin-protein interactions, these interactions are the most studied polyphenol-protein interactions. Some studies have used bovine serum albumin, while others used 𝛼-lactalbumin. 188 The storage of polyphenols for a long time under specific environmental conditions such as high temperature and light exposure could seriously affect their chemical and physical stabilities. With respect to long-term storage, two phenomena must be considered in food systems: (i) the polyphenol components may be altered by oxidation, and (ii) the physical structure of polyphenols may be influenced by polymerization. Normally, oxidative damage is hardly a problem in practice, but it can be minimized by protection from light and air using an inert atmosphere as a preventive measure to maintain the effective biological activity of compounds.

Reducing sugars 189 or different carbohydrates 190 (e.g. trehalose) are commonly used as antioxidants in foods. Komes et al. 190 showed that due to its glass transition property, trehalose can retain and preserve hydrophobic phenolic compounds of fresh fruits during dehydration processes.

Other phytochemicals and antioxidants intentionally introduced to a food system may help to stabilize polyphenols. [START_REF] Negi | Plant extracts for the control of bacterial growth: efficacy, stability, and safety issues for food application[END_REF] Vitamin C added to processed yellow passion fruit exerted a protective effect on plant chemicals. 191 Red clover leaf extracts were demonstrated to make anthocyanins more stable when added to muscadine wines during storage (20 and 37 ∘ C for 9 weeks). 192 Lecithin addition to tea catechin solution at acidic pH and room temperature has the ability to protect tea catechins from oxidative damage. 193 High-performance liquid chromatography (HPLC) coupled with mass spectrometry (MS) analysis is usually used to assess the stability of polyphenols (e.g. molecular structure and quantity). 194 Quantitative structure-activity relationship (QSAR) studies can be used to predict and determine the extract formulations that would be more stable in different food matrices. Indeed, the QSAR has already been used to determine the structure-reactivity and structure-antimicrobial activity relationships of phenols under different conditions by evaluating different parameters. 195 In this way, QSAR analysis can provide information on how interactions between phenolics or with other molecules (proteins, lipids, oxygen, etc.) modulate their antimicrobial activity.

Release of plant polyphenols from active edible coatings or packaging materials to foods in direct contact

To alleviate the deficiencies in using plant phenolic compounds in foods, polyphenols can be added to the immediate zone of foods in direct contact through slow release from edible coatings or packaging materials. Such systems could maintain an efficient concentration of antimicrobial plant phenolics in the superficial zone of foods over time. This approach could be advantageous for foods such as raw muscle foods (fish fillets, meat pieces) or some fruit and vegetables for which most microbial contamination occurs in their superficial zone. One advantage of edible coatings or food-contact packaging materials with antimicrobial plant phenolics over their direct spraying is their controlled release over time. Various studies on the use of active films and edible coatings to deliver antimicrobial agents to the surface of a wide range of foods in contact (fruit, vegetables, and meat products) have been conducted. 196,197 The delivery of phenolic compounds from edible coatings is mainly described as a sequence of material transfer movement starting with diffusion, followed by desorption from the film's or coating's surface, sorption of the compounds at the interface and finally sorption into the food. [198][199][200][201][202][203] It has been acknowledged that the delivery rate of bioactive compounds from films or edible coatings to food is faster when the release is a consequence of their swelling or dissolution, which is conditioned by the nature of the food matrix in direct contact and the polymer matrix of film or coating. 200 Furthermore, the time and temperature of contact, 204 the polymer matrix (promoting or no interaction) with phenolics via functional groups, 199,201,203 the properties (chemical structure and polarity) 200 of the phenolic compounds (migrating substances), and their contents in the edible films have been reported to affect the migration rate through coatings to foods in contact. 198,199,202,205 Additionally, the microstructure of the polymer matrix 206 and the way in which the phenolic compounds are oriented with respect to the food based on their hydrophilic or hydrophobic properties strongly affect their migration and thus their effectiveness in protecting foods in contact. Edible coating or active packaging design should thus exploit the possibility of tuning the physicochemical interactions between antimicrobial plant phenolics and the polymers, which are the main components of edible coatings/packaging materials, to control their release kinetics. Another potential advantage of edible coatings/packaging materials incorporating antimicrobial plant phenolics is their increased stability to oxidation in these polymeric matrices. Packaging materials made of edible biopolymers combined with natural antimicrobials are favorable for making foods safer and of higher quality. 201 The association of antimicrobial agents with edible coatings has thus increasingly been considered a favorable approach to increase the shelf life and/or enhance the safety of perishable foods in recent decades. This trend is illustrated by the increase in the proportion of articles on edible coatings that also consider antimicrobials in the Web of Science ® database from 0% before 1994 to 25% since 2010.

Edible coatings/films incorporating antimicrobial plant polyphenols

Edible coating/film formulations are based on only food ingredients and additives. On the one hand, an edible coating is described as a thin layer of edible material formed directly on the superficial zone of a food that can be consumed with the food product. To prepare edible coatings, film-forming suspensions can be applied to foods by various processes, as reviewed by Andrade et al. 207 (e.g. panning,fluidized bed,dipping,spraying). A subsequent draining of excess film-forming suspension and drying are necessary following the dipping of food products. On the other hand, stand-alone edible films may be prepared by either solvent-casting or extrusion-blowing methods. Cast films can be prepared by pouring a film-forming suspension on a flat substratum, which is subsequently dried. Most edible films are prepared using this versatile technology at the laboratory scale. However, at the industrial scale, most food packaging plastic films are prepared by extrusion-blow molding. Edible films can also be prepared with this technology. 208 Edible coatings/films provide a barrier to gaseous exchange as well as the transmission of moisture, flavors and other soluble constituents of processed products when manipulated and stored, thereby enhancing their shelf life. 209 The use of edible coatings/films as vectors of bioactive molecules can ensure their availability to act effectively at their site of action. 210 Incorporation into edible coatings/films is a good alternative to preserve bioactive compounds such as antimicrobial phenolics in foods. Bioactive compounds can be incorporated (i) on the external surface of the film, (ii) on the internal surface of the film, (iii) in the multilayers of the edible coatings, or (iv) in different parts of the film. [START_REF] Quirós-Sauceda | Edible coatings as encapsulating matrices for bioactive compounds: a review[END_REF] Zein, whey proteins, caseinates, soy proteins, chitosan, alginate, carrageenan, pullulan, pectin, cellulose, and its derivatives are examples of biopolymers that have been used to prepare edible films and coatings. [211][212][213][214][215][216][217][218][219] These coatings and films can delay food spoilage due to their gas barrier properties, their intrinsic activity (e.g. the antifungal activity of chitosan) or the antimicrobial compounds added to their formulation. Therefore, in addition to the biodegradability of these natural biopolymers, such films and coatings are considered environmentally friendly since they could contribute to food waste reduction.

Recent studies dealing with the incorporation of antimicrobial plant extracts into edible films and coatings are listed in Table 3. The incorporation of such compounds into films may enhance their antimicrobial activity; 231 the uptake of a sufficient quantity of phytochemicals on the surface of foods over time could be made possible by their gradual release through the film. [START_REF] Knobloch | Progress in Essential Oil Research[END_REF] In the literature, antimicrobial packaging acts primarily on foodborne pathogenic bacteria such as Listeria monocytogenes, Staphylococcus aureus, Escherichia coli O157:H7, and Salmonella spp. [232][233][234] However, food spoilage microorganisms such as Bacillus spp. and Lactobacillus spp. could also be targeted 228 to increase perishable food shelf life and/or contribute to food waste reduction.

The release of antimicrobials from the edible coatings depends on many attributes, such as electrostatic (ionic and hydrogen bonds) and hydrophobic interactions between antimicrobials and polymers, osmosis, structural modifications of the polymeric matrix resulting from the presence of the antimicrobial agents, and the surrounding conditions (temperature, pH). [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] Coatings may be used as a system for releasing antimicrobial phytochemicals over time on a wide range of food surfaces (e.g. vegetables, fruit, and meat products). [START_REF] Da | Application of plant derived compounds to control fungal spoilage and mycotoxin production in foods[END_REF] Rapeseed protein/gelatin coatings containing grape seed extract have been used to inhibit Escherichia coli O157:H7 and Listeria monocytogenes growth in strawberries. 235 Edible zein coatings applied to fruit and vegetables have been found to have a protective effect during storage (by controlling their respiration, ripening, and senescence). 201 Some phenolic acids (gallic, vanillic, and cinnamic acids) and extracts from clove, oregano, artichoke stems and walnut shells were assessed as antimicrobial zein film additives against four plant pathogenic bacteria. 201 The incorporation of phenolic compounds (10-40 g m -2 ) into zein films was shown to improve the porosity of films, which became more flexible and lost their brittleness.

Chitosan-coated films incorporated with green tea extract (40 g kg -1 ) had bactericidal activity against Listeria monocytogenes in ham steak for 8 weeks of storage at 4 ∘ C. 236 In a more recent study, incorporation of green tea extract (5-10 g L -1 ) considerably enhanced the antifungal activity of chitosan coatings on fresh walnut kernel; fungal growth was not detected during 18 weeks of storage at room temperature. 237 Food packaging materials incorporating antimicrobial plant polyphenols Although conventional plastic packaging raises environmental concerns and its current use in direct contact with foods is highly regulated, many studies regarding active food packaging consider polyethylene or polypropylene and their derivatives due to their excellent physical and chemical characteristics. 161,238,239 Such active packaging falls under the scope of the EC 450/2009 regulation. 240 At the industrial scale, there are two possible processes for including antimicrobials in packaging materials: (i) direct incorporation into the polymers during extrusion by melt-blending 241 and (ii) coating of the antimicrobial agents onto polymer surfaces. 242,243 Extrusion is preferred by manufacturers because of the high cost of the coating process (additional steps and technical changes). 244 However, additional costs may also result from the degradation of actives during extrusion processes operating at high temperature, which would have to be compensated for by the addition of a higher amount of active ingredients. 241 Moreover, the fact that antimicrobials are equally spread throughout the entire thickness of films made by extrusion can lead to limitations in their release from the packaged material and therefore lower in situ activity. 241 Therefore, in addition to the question of the stability of antimicrobial plant phenolics to the conditions prevailing during melt blending with the polymer (high temperature and shear stress), the transport properties of the plant phenolics once incorporated into the polymeric matrix of the packaging material also must be considered. The amount of plant phenolics released and their release kinetics should effectively inhibit the multiplication of unwanted microorganisms in the superficial zone of food.

Adequate analytical methods to assay plant phenolics in active food packaging materials and monitor their release in food matrices in direct contact should thus be developed. Therefore, Colon and Nerin 245 developed a method to quantify tea compounds released from a polyethylene terephthalate (PET) film with an internal coating layer containing green tea extract to IV gamma nectarines placed in a tray covered by this active film. Solid-phase extraction (SPE) combined with ultra-performance liquid chromatography (UPLC)-MS was used to check that the amounts of catechins and caffeine from the green tea extract delivered by the packaging material were below the migration limits of 10 μg per kg of nectarine. In agreement with the European Union (EU) regulations for food-contact materials (EU 10/2011 [START_REF]99 Food and Drug Administration (FDA)[END_REF] and EC 450/2009 240 ), all components not available in the positive lists of the regulations introduced and not specifically recognized as a food additive, such as catechins and caffeine, must be below 10 μg per kg of food or food simulant. The ability of packaging films composed of cast polypropylene/polyvinyl alcohol incorporated with rhubarb ethanolic extracts and cinnamon essential oil to preserve the quality of fresh beef was investigated. Interestingly, all the experimental cast films used significantly decreased the total viable counts (TVCs). 246 However, as stated earlier, preparation of cast films by solvent evaporation, despite likely limiting essential oil loss by evaporation during the preparation of stand-alone films, is not the preferred method. In another study, 15 g kg -1 pomegranate peel extract was blended with polyethylene resin by twin-screw extrusion at 160-190 ∘ C to obtain active films. 161 A decrease in total volatile basic nitrogen (TVB-N) was observed during refrigerated storage of fresh pork meat packaged in these active films. The shelf life of the fresh pork meat was thus extended by 3 days. Interestingly, this result suggests that pomegranate peel extract still had significant antimicrobial and antioxidant activities following its incorporation by extrusion in polyethylene films. Despite its promise, this type of practice is still in its infancy. 247,248

CONCLUSION

The in vitro antibacterial activity of many phenolic-rich plant extracts and pure plant phenolics has been reported in this review. Although their activity in perishable foods is often reduced or even lost, numerous examples of plant extracts or phenolics effectively preventing microbial contamination or degradation of foods are given. Since no pure plant phenolics are authorized as food preservatives, only direct incorporation of edible plant extracts (i.e. food ingredients) into perishable foods can be considered currently. In addition to direct incorporation into perishable food matrices, incorporation into the polymeric matrix of either food packaging materials or edible coatings is a promising approach to deliver active plant phenolics to the immediate zone of foods in direct contact. However, again, due to the absence of active plant phenolics from the positive list of food preservatives and the legislation regarding active food packaging, only the application of edible coatings made of biopolymers with a food ingredient or additive status and edible plant extracts with a food ingredient status can be considered today. In addition to evaluation of some plant phenolics as new food preservatives, future research on building a rationale for the application of phenolic-rich plant extracts or phenolics for food preservation should specifically focus on (i) identifying the molecular mechanisms underlying their ability to control unwanted microorganisms, (ii) understanding the effect of food microstructure and composition on their antimicrobial activity and (iii) designing innovative and sustainable systems of delivery of active phenolics preserving their stability before use and favoring their controlled release in the superficial zone of perishable foods where postprocessing microbial contamination mainly occurs.

Figure 2 .

 2 Figure 2. Different sites of action of antimicrobial polyphenols at the cellular level.

ACKNOWLEDGEMENT

The authors gratefully acknowledge the French National Agency for Research (ANR-14-CE20-0005-01 ACTIPHEN) for the financial support of this work.

DECLARATION OF INTEREST STATEMENT

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

AUTHOR CONTRIBUTIONS

All authors listed have made substantial, direct and intellectual contribution to the work and approved it for publication.


U Tan
email: u.tan@laposte.net

‡ O Rabaste

C Adnet

J.-P Ovarlez

A Sequence-Filter Joint Optimization

Keywords: Gradient descent, mismatched filter, optimisation methods, waveform design

This paper deals with the optimisation of a sequence and its associated mismatched filter. This question has already been addressed in the literature, by alternatively solving two minimisation problem, one per sequence, meaning that the optimisation is never performed on both sequences at the same time. So, this article introduces some methods in order to optimise jointly (i.e. simultaneously) a sequence and its filter. First, a gradient descent is basically applied on both sequences. Second, another algorithm studies an objective function that includes the optimal mismatched filter that minimises the Integrated Sidelobe Level (or the Peak-to-Sidelobe Level Ratio). Simulations show promising results in terms of sidelobes: as expected, a joint optimisation seems to perform better than a separate one. As both methods behave differently, their choice will depend on the applications.

I. INTRODUCTION

In radar processing, the received signal is often correlated with a filter to detect the presence of a target. If this filter is the transmitted signal, it is identified as a matched filter, and otherwise as a mismatched filter.

The matched filter is known for maximizing the SNR (Signal-to-Noise Ratio) in Gaussian Noise, but it may suffer from high sidelobes. These sidelobes can be reduced with a mismatched filter, at a cost of some loss in processing gain. In both cases, their design can be done considering an optimisation problem, in which the cost function is a criterion such as the SINR (Signal-to-Interference plus Noise Ratio), the ISL (Integrated Sidelobe Level) or the PSLR (Peak-to-Sidelobe Level Ratio).

optimising the matched filter output is a quite complicated task, especially when the transmitted sequence is subject to the constant modulus constraint, in which case the optimisation problem is not convex. Thus, a lot of methods have been reviewed in the literature [START_REF] Tan | Optimization methods for solving the low autocorrelation sidelobes problem[END_REF]. Stochastic methods are usually helpful, because their convergence is almost certain theoretically (in the sense that, one day, a basic stochastic algorithm will converge to the global optimum) [START_REF] Deng | Polyphase code design for orthogonal netted radar systems[END_REF]. But due to the large dimension of the variables to optimise, they might not be appropriate here. Among deterministic algorithms, the gradient descent has shown interesting capabilities in similar problems [START_REF] Baden | Efficient energy gradient calculations for binary and polyphase sequences[END_REF]. Even if it converges to a local minimum -or to a saddle point -solutions obtained with this strategy present remarkable sidelobes.

On the other hand, a mismatched filter is not subject to the constant modulus constraint, so that the corresponding optimisation problem is much simpler. It has indeed been shown in [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF] and in [START_REF] Stoica | Optimization of the receive filter and transmit sequence for active sensing[END_REF] that an optimal mismatched filter that minimises the ISL or the PSLR can be calculated through a convex problem. For instance, in [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF], the mismatched optimisation problem is reformulated as an equivalent convex quadratically constrained quadratic program (QCQP). It may thus be possible to find a global minimum, so-to-speak an optimal mismatched filter for a given sequence.

More recently, several approaches have been introduced in order to optimise the pair transmitted sequence/mismatched filter in a sequential way, alternatively solving two optimisation problems, one per sequence. For instance, Karbasi et al. [START_REF] Karbasi | Design of multiple-input multiple-output transmit waveform and receive filter for extended target detection[END_REF] considers a semi-definite programming (SDP) relaxation of the constraint. While in [START_REF] Stoica | Optimization of the receive filter and transmit sequence for active sensing[END_REF], a cyclic algorithm is used to study the mean-square error of the estimation of the SINR in the frequency domain.

However, all these algorithms are cyclic, meaning that the optimisation is never performed on both sequences at the same time. Hence, this article deals with a joint design of a sequence and its associated mismatched filter. Two methods are proposed here. The first method corresponds to a classic gradient descent, that will operate on both sequences at the same time, like in [START_REF] Xu | Joint design of phase coded waveform and mismatched filter[END_REF]. However, constraints on the loss-inprocessing gain (due to the usage of a mismatched filter) are introduced as a pattern. The second proposed algorithm exploits the existence of the optimal mismatched filter in the ISL or the PSLR sense. This optimal solution will be inserted into the objective function, producing a signal-only optimisation problem. This new cost function will also be minimised with a gradient descent.

This article is organised as follows. Section II gives some reminders on the matched filter and the mismatched filter. In Section III, two methods are introduced for optimising jointly a sequence and its associated mismatched filter: a direct solution, based on a gradient descent, and an indirect one that exploits the existence of an optimal mismatched filter under a criterion. Some results are presented in Section IV, according to different initialisations.

Notation: In the following, bold letters designate matrices and vectors. (.) * and (.) H denote the conjugate and the transpose conjugate operator, respectively. • and * designate the Hadamard and the convolution product. (.) r denotes the reverse operator: for a given vector a := [a 1 , . . . , a N ], a r := [a N , . . . , a 1 ]. . 2 stands for the Frobenius norm. For an m × n matrix A, it is defined by

A 2 2 = m i=1 n j=1 |a i,j | 2 .
While A ∞ = max i,j |a i,j |.

II. REMINDER ON THE MATCHED FILTER AND THE MISMATCHED FILTER

This section reminds some definitions on the matched filter and the mismatched filter. It also explains how an optimal mismatched filter can be obtained through an optimisation problem, for a given sequence. More details are given in [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF].

A. Definitions

Let s be a discrete signal containing N samples:

s = [s 1 , s 2 , . . . , s N ] T . (1)
In the following developments, sequence s belongs to the Ndimensional torus (or hypertorus), denoted T N , which is a non-convex set. In other words, s is subject to the constant modulus constraint. Let α k ∈ [0, 2π[ be the phase angle of the element s k :

s k = exp(jα k )/ √ N if k ∈ 1, N 0 otherwise. ( 2
)
Matched filtering consists in a comparison of the signal s and a time-shifted version of itself, i.e., generating sequence y of length 2N -1 such that:

y = Λ N (s)s * , (3)
where Λ K (s) is a matrix of size K + N -1 × K containing delayed versions of the sequence s, such that:

Λ K (s) :=                            s N 0 • • • • • • • • • • • • 0 . . . s N . . . . . . s 2 . . . 0 . . . s 1 s 2 • • • s N 0 • • • 0 0 s 1 . . . . . . s N . . . . . . . . . . . . . . . s 2 . . . 0 0 • • • 0 s 1 s 2 s N . . . 0 s 1 . . . . . . . . . . . . . . . s 2 0 • • • • • • • • • • • • 0 s 1                            . (4)
On the other hand, processing signal s with a different filter q of length K is called mismatched filtering:

y = Λ K (s)q.
(5)

A mismatched filter q is less constrained than the matched filter, since it can take any value in C K . Moreover, its length may also differ from N , and can be in particular chosen to contain more elements, thus providing additional degrees of freedom compared to the matched filter. It will be assumed without loss of generality that K = N + 2p, p ∈ N, so that the length of y is odd.

The matched filter is known for maximizing the SNR (Signal-to-Noise Ratio) at the peak response. Using a mismatched filter implies inevitably a loss-in-processing gain (LPG), expressed by (under a white noise hypothesis):

LPG = 10 log 10 SNR mismatched SNR matched = 10 log 10 |q H s| 2 (q H q)(s H s) ≤ 0. (6)
This loss-in-processing gain can be inserted as a convex constraint in optimisation problems, depending on s and q, as noticed in [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]:

LPG ≥ 10 log 10 (α) =⇒ q H q ≤ αs H s. (7)

B. Optimal Mismatched Filters

Several criteria have been introduced in order to measure the performance of these filters: the Merit Factor [START_REF] Golay | Sieves for low autocorrelation binary sequences[END_REF], the Integrated Sidelobe Level (ISL) or the Peak-to-Sidelobe Level Ratio (PSLR).

1) Integrated Sidelobe Level: The ISL is defined by:

T N × C K → R+ (s, q) → ISL(s, q) := y H F y, ( 8
)
where y is defined as in ( 5) and F is a diagonal matrix of order K + N -1, defined by the vector [1, . . . , 1, 0, 1, . . . , 1], with ones except for a 0 at the entry N + p.

The ISL can be considered as an objective function of an optimisation problem, called in this article (P ISL ):

(P ISL ) min q ISL(s, q) s.t. s H q = s H s, (9)
that can be solved analytically, using Lagrangian multipliers [START_REF] Griep | Poly-phase codes and optimal filters for multiple user ranging[END_REF]:

q ISL (s) = s H s Λ K (s) H F Λ K (s) -1 s s H (Λ K (s) H F Λ K (s)) -1 s . ( 10
)
This last definition means that, for a given sequence s, there exists a unique and optimal filter q ISL (s) that minimises the ISL. Note here that this analytic solution does not guarantee an acceptable loss-in-processing gain. If the LPG is added as a second constraint in the constrained optimisation problem (P ISL ), then the new problem cannot be solved analytically anymore to our knowledge.

2) Peak-to-Sidelobe Level Ratio: The PSLR is expressed by:

T N × C K → R+ (s, q) → PSLR(s, q) := F y 2 ∞ |s H q| . ( 11
)
Similarly, consider the following optimisation problem:

min q PSLR(s, q) s.t. s H q = s H s. ( 12
)
This optimisation problem is equivalent to the following QCQP (Quadratically Constrained Quadratic Program) [START_REF] Rabaste | Mismatched filter optimization for radar applications using quadratically constrained quadratic programs[END_REF]:

(P PSLR )        min q,t t s.t. s H q = s H s q H λ H N +p+i (s)λ N +p+i (s)q ≤ t for |i| ≥ 1, (13)
where λ k (s) is the k-th row of the matrix Λ K (s). The previous optimisation problem is convex, because the cost function and all the constraints are. Hence, a global solution can be computed, denoted here by q PSLR (s), using any convex optimisation method such as the interior point method for instance [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.1[END_REF]. Note here that since the constraint on the LPG is quadratic and convex, it can easily be added to the optimisation problem (P PSLR ), the resulting problem still providing a global solution.

III. JOINT OPTIMISATION ALGORITHMS

The previous section pointed out that, for a given sequence, there exists an optimal mismatched filter, whatever norm is considered. As mentioned in the introduction, some papers (see [START_REF] Stoica | Optimization of the receive filter and transmit sequence for active sensing[END_REF], [START_REF] Karbasi | Design of multiple-input multiple-output transmit waveform and receive filter for extended target detection[END_REF] for instance) have dealt with an optimisation of the sequence and its associated mismatched filter, but in a sequential way: two optimisation problems are considered, one for each sequence (while the other is set as a constant), see Table I. It enables to apply the property of existence of an optimal mismatched filter, but it does not imply at all that the obtained pair is optimal.

Hence, in this section, two algorithms are introduced in order to optimise jointly a sequence and its associated mismatched filter. First, a gradient descent is applied on both sequences, on a cost function that includes a pattern. This pattern allows to control in particular the loss-in-processing gain and the sidelobe level. Second, a modified version of the cost function is considered. It exploits the existence of an optimal mismatched filter, in the PSLR or the ISL sense. A constraint can be added in order to adjust the LPG.

A. Direct Joint Optimisation

This section proposes to jointly optimise a transmitted sequence and its associated mismatched filter, via a gradient descent on s and q. A similar method has already been proposed in [START_REF] Xu | Joint design of phase coded waveform and mismatched filter[END_REF] using the PSLR as a cost function: however, constraints are managed through some penalty coefficients that have to be determined empirically. Here, the sidelobe level and the loss-in-processing gain are handled through a userdefined objective pattern, inserted in the cost function. It seems reasonable to put these sidelobes as low as possible but, in practice, the convergence speed is affected. It is in fact more reliable to consider a feasible pattern an algorithm can reached, and then adjust the former (e.g., a reduction of the sidelobe level) if necessary.

As said in Section II, for a sequence s of length N and its associated mismatched filter q of length N +2p, the processing output is of length 2N + 2p -1. And so should be the pattern, here denoted by g. An example of pattern is given in Fig. 1. The modified optimisation problem, that includes the pattern, is the following:

       min s,q E 1 (s, q) := 2N +2p-1 k=1 |y k /g k | 2ρ -1 2 , s ∈ T N , q ∈ C K , (14)
and where y is defined as in Section II. This optimisation problem searches for couple (s, q) that provides the closest output to the pattern.

In order to apply a gradient descent, a gradient computation of the cost function is needed, which can be done numerically with finite differences. Otherwise, for sequence s, notice that the considered partial derivatives are computed with respect to the phase α k of each element s k , so that the constant modulus property is respected:

∂E 1 ∂α k = -2ρ [s k ((γ • y) * q) k ], k ∈ 1, N (15)
where

γ k = (|y k /g k | 2ρ -1) |y k /g k | 2ρ-1 (1/g 2 k )
. While for the mismatched filter, Wirtinger derivatives are used:

∂E 1 ∂q k = ρ ((γ • y) * s r ) , k ∈ 1, 2N + 2p -1 (16)
After convergence of the steepest descent step, the pattern may be slightly adjusted (reduction of 1 dB of the sidelobes for instance). The whole procedure is summarized in Table II.

B. Indirect Joint Optimisation

An iterative algorithm is also proposed in this section. It takes advantage of the existence of an optimal mismatched filter that minimises the PSLR or the ISL, as explained in Section II. Only the latter is detailed in this section, even though the following procedure can easily be extended for the PSLR.

For a joint optimisation, the optimisation problem (P ISL ) becomes:

(P (2) ISL )        min s,q E 2 (s, q) := F Λ K (s)q 2 2 s.t. s H q = s H s s ∈ T N , q ∈ C K . (17)
Notice that the only difference is on the optimisation variables, with the addition of s. Since sequence s belongs to an hypertorus, which is a non-convex set, this problem cannot be freely solved globally, contrary to the original problem (P ISL ).

Separating both optimisation variables gives the following problem:

(P (3) ISL )        min s min q E 2 (s, q) = F Λ K (s)q 2 2 s.t. s H q = s H s s ∈ T N , q ∈ C K . (18)
In this expression, (P ISL ) can be identified. As mentioned earlier, there exists a global solution that can be computed, denoted q ISL (s) -its explicit defintion is defined in Section II. Remark that q ISL (s) can also be computed through a QCQP. This solution is put into the objective function, becoming a function of a sequence-only variable:

(P (4) ISL ) min s E 3 (s) := E 2 (s, q ISL (s)) s ∈ T N . ( 19
)
As such, the definition of the mismatched filter is therefore inherent, given the expression of cost function E 3 . At the end of the day, the proposed algorithm is actually akin to a simple minimisation problem, that can be solved using a gradient descent for instance. Notice that a numerical computation of the derivative requires to consider an optimisation problem (P ISL ) for each phase α k ... The algorithm is summarized in Table III. Remind that it can easily be extended to an optimisation of the PSLR criterion, and/or with constraints on the loss-in-processing gain.

IV. RESULTS

This section presents some results obtained with the previous described algorithms. Drawn sequences are of length N = 64. Different initialisations are compared.

To solve all the QCQP, the CVX package has been used. CVX is a package for specifying and solving convex programs [START_REF] Grant | CVX: Matlab software for disciplined convex programming, version 2.1[END_REF].

In the following results, different pairs sequence/filter have been considered, obtained through different strategies. They will denoted as follows:

• "Matched opt.": an optimisation of the matched filter has been performed, like in [START_REF] Baden | Efficient energy gradient calculations for binary and polyphase sequences[END_REF]. • "Direct opt.": a direct joint optimisation with a gradient descent, described in Section III-A. • "Indirect opt.": an indirect joint optimisation, proposed in III-B.

A. First Results

Figure 2 illustrates the usefulness of a joint optimisation. It compares a sequential algorithm with a joint one, proposed in the previous section. Therefore, we have first considered a random sequence, optimised it in the sense of a matched filter, and then computed its optimal mismatched filter in the PSLR sense. Second we have used this sequence as the initialisation sequence for the proposed joint optimisation algorithm. Whatever method, the pair sequence/mismatched filter provided by the proposed joint optimisation algorithm outperforms the results provided by the separated optimisation, of around 10 dB.

Given

Initial sequence s Initial mismatched filter q Repeat 1. Gradient descent search on s, q fixed (see below) 2. Computation of q PSLR (s), optimal solution of (P PSLR ) Until A stopping criterion is satisfied.

s k = s k e -jµ(∇αE 1 ) k , k ∈ 1, N -Update of q : q k = q k -µ(∇qE 1 ) k ), k ∈ 1, K -Repeat until convergence 2.

Readjustment of the pattern g Until

A stopping criterion is satisfied.

Table III AN INDIRECT JOINT OPTIMISATION ALGORITHM

Algorithm Indirect joint optimisation based on a gradient descent

Given

Initial sequence s Repeat 1. Gradient descent search -Computation of the gradient vector ∇αE 3 -Search of the best step µ -Update of s :

s k = s k e -jµ(∇αE 3 ) k , k ∈ 1, N Until
A stopping criterion is satisfied.

A comparison of both optimisation methods described in this article is represented in Figure 3. A loss-in-processing gain of 1 dB has been allowed. In terms of PSLR, a direct gradient descent seems more appropriate (PSLR = -58 dB). However, note that this pattern-based method cannot easily been adapted to provide good ISL sequences. In that aspect, the second proposed method provides very interesting performance in terms of ISL, with very low sidelobes near the mainlobe, at the expense of relatively higher sidelobes at the end of the sequences (yet at a very acceptable level, around -40 dB, at most).

B. On the Influence of the Initialisation

It has already been mentioned that, theoretically, a gradient descent converges to the closest local minimum (at least to the closest saddle point). In this regard, the initialisation is quite important, because it influences the sidelobe level of the obtained pair sequence/filter. Two initialisations are thus compared in this section: the first one was a random initialisation (see the previous paragraph), while the second one uses as initialisation a sequence already optimised in the PSLR sense. Results are presented Figure 4.

Starting from the random sequence drawn in Section IV-A, an optimisation of its matched filter is performed, such as in [START_REF] Baden | Efficient energy gradient calculations for binary and polyphase sequences[END_REF]. The obtained sequence is used as an initialisation for both proposed methods. In addition to low sidelobes, it is interesting to look at their distributions, according to the method. A direct optimisation of the PSLR gives -as expected -a uniform distribution, with sidelobes at -58 dB, improved by 8 dB with the other init. On the other hand, the indirect optimisation over the ISL provides incredibly good results, with sidelobes around the mainlobe reaching levels as low as -90 dB while the sidelobes at the edges are still acceptable, around -50 dB. To our knowledge, such sidelobes level have not been reached in the literature. The particular shape obtained in this ISL case may besides be very interesting in radar applications, as it may improve the detection of close and small targets that are usually hidden in strong clutter.

V. CONCLUSION

In this article, several algorithms have been introduced, in order to design jointly a sequence and its mismatched filter. Unlike methods described in the literature, the optimisation process is performed simultaneously on both sequences, and not alternatively. Simulations have highlighted really promising results on the sidelobes, better than cyclic algorithms:

• In terms of sidelobe levels, obtained results are quite satisfying, even with small sequences. • According to the algorithm, their distribution is different.

A joint direct optimisation of the PSLR provides -as expected, because of the chosen pattern -a mismatched filter output with a flat profile. While an indirect joint optimisation over the ISL shows amazingly low levels around the mainlobe. Ongoing works will be focused on:

• applying other methods instead of a gradient descent;

• the initialisation choice for the mismatched filter;

• an extension of these algorithms for a set of sequences (and their associated set of mismatched filters); • comparing output levels with some lower bounds on the correlation sidelobe level of sets of sequences, such as Welch or Levenshtein bounds. Note that, for a single sequence, these bounds are equal to 0 (-∞ dB), which leaves some possibility to reach even lower sidelobe levels with joint optimisation methods.

Figure 1 .

 1 Figure 1. Some patterns. An amplified loss of 5 dB is authorized.

Figure 2 .

 2 Figure 2. Separated Optimisation vs. Joint Optimisation

Figure 3 .

 3 Figure 3. Comparison of Joint Optimisation Methods

Figure 4 .

 4 Figure 4. Joint Optimisation, Starting from a "Flat" Sequence

Table I A

 I CYCLIC ALGORITHMAlgorithm Cyclic optimisation with a gradient descent

ACKNOWLEDGEMENT

This work was funded by Thales Land & Air Systems and was partly supported by a DGA-MRIS scholarship.