F Marquier

C Arnold

M Laroche

J J Greffet

Y Chen

Degree of polarization of thermal light emitted by gratings supporting surface waves

Absorption and emission of light due to the resonant excitation of surface waves on a grating is a well-known phenomenon. We report the first complete study of the influence of the role of angle and polarization on thermal emission by lamellar gratings. We derive the emitted Stokes vectors in any direction. We find that a source can be quasi isotropic from the point of view of the intensity but strongly anisotropic for polarized light. It follows that the degree of polarization can vary between 0 and 1, depending on directions.

Introduction

The possibility of engineering the thermal emission properties of materials by taking advantage of surface resonances has been demonstrated in the early nineties using doped silicon gratings [START_REF] Hesketh | Organ pipe radiant modes of periodic micromachined silicon surfaces[END_REF]. The authors reported important modifications of the emissivity depending on the surface of the material and also on the polarization of the detected wave. Further experiments taking advantage of surface plasmons have been performed on a gold grating for which thermal emission has been shown to be directional at a given wavelength [START_REF] Kreiter | Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons[END_REF]. Actually, it has been known since the seventies that 100% absorption can be obtained with a metallic grating [START_REF] Wood | On a remarkable case of uneven distribution of light in a diffraction grating spectrum[END_REF][START_REF] Maystre | The total absorption of light by a diffraction grating[END_REF] for given angle of incidence and wavelength corresponding to the resonant excitation of a surface plasmon. According to Kirchhoff's law [START_REF] Greffet | Field theory for the generalized bidirectional reflectivity: derivation of Helmholtz's reciprocity principle and Kirchhoff's law[END_REF], the emissivity is then 100%: this has been observed in ref [START_REF] Kreiter | Thermally induced emission of light from a metallic diffraction grating, mediated by surface plasmons[END_REF][START_REF] Laroche | Highly directional radiation generated by a tungsten thermal source[END_REF][START_REF] Heinzel | Radiation filters and emitters for the NIR based on periodically structured metal surfaces[END_REF]. To be more general, total absorption or emission can be obtained when coupling a surface wave to a propagating wave in free space. Such a surface wave can be either a surface plasmon-polariton for metals and doped semiconductors or a surface phonon-polariton for polar materials, or a surface wave along a truncated photonic crystal [START_REF] Laroche | Coherent thermal antenna using a photonic crystal slab[END_REF]. Experiments using infrared surface phonon polaritons propagating along SiC gratings were reported in ref. [9][START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF][START_REF] Greffet | Coherent emission of light by thermal sources[END_REF]. The authors showed that a SiC grating can emit very narrow peaks for a given wavelength: the grating behaves like a thermal antenna. It was stressed that this property is a consequence of the spatial coherence of the fields in the source due to the presence of the surface wave. It has been shown that directional emissivity is related to the propagation length of the surface wave which appears as a spatial coherence length. Therefore this effect has been called coherent thermal emission [START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF][START_REF] Greffet | Coherent emission of light by thermal sources[END_REF].

Many other sources have been proposed to achieve coherent thermal emission using periodic structures, cavities [START_REF] Celanovic | Resonant-cavity enhanced thermal emission[END_REF][START_REF] Battula | Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance[END_REF] or geometrical resonances [START_REF] Laroche | Tuning the optical response of nanocylinder arrays: An analytical study[END_REF]. It has been shown that a SiC wafer surrounded by a planar multilayer structure exhibits strong directional emission peaks [START_REF] Lee | Design and fabrication of planar multilayer structures with coherent thermal emission characteristics[END_REF]16]. In such a system, evanescent waves exist in the stop band of the photonic crystal, which can be coupled to surface phonon polariton. Hence, it is possible to couple surface waves to propagating waves and generate directional thermal emission. Multilayers structures can also be used as Bragg's mirrors to create a resonant cavity and directional emission [START_REF] Celanovic | Resonant-cavity enhanced thermal emission[END_REF]. Battula and Chen working with both cavity and surface plasmons modes also obtain very directional thermal emission [START_REF] Battula | Monochromatic polarized coherent emitter enhanced by surface plasmons and a cavity resonance[END_REF]. Waveguide modes can also produce directional emission as demonstrated in ref. [START_REF] Joulain | Coherent thermal emission by microstructured waveguides[END_REF]. The application to spectrally selective surfaces has been considered by some authors [START_REF] Enoch | Simple layer-by-layer photonic crystal for the control of thermal emission[END_REF][19][20][START_REF] Sai | Numerical study on spectral properties of tungsten one-dimensional surface-relief gratings for spectrally selective devices[END_REF][START_REF] Hu | Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation[END_REF]. In the previous papers, only one-dimensional structures (1D-photonic crystal or lamellar gratings) were used [9][START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF][START_REF] Greffet | Coherent emission of light by thermal sources[END_REF][START_REF] Laroche | Tailoring silicon radiative properties[END_REF]. Two dimensional periodic structures have also been used to couple propagating waves to a resonant mode [START_REF] Hu | Effects of Periodic Structures on the Coherence Properties of Blackbody Radiation[END_REF][START_REF] Pralle | Photonic crystal enhanced narrow-band infrared emitters[END_REF][START_REF] Maruyama | Thermal radiation from two-dimensionally confined modes in microcavities[END_REF][START_REF] Chan | Thermal emission and design in 2D-periodic metallic photonic crystal slabs[END_REF]. The emission properties of 3D-photonic crystals have been studied since 1999 by several authors [START_REF] Heinzel | Radiation filters and emitters for the NIR based on periodically structured metal surfaces[END_REF][START_REF] Chan | Direct Calculation of thermal emission for three-dimensionally periodic photonic crystal slabs[END_REF][START_REF] Yannopapas | Thermal emission from three-dimensional arrays of gold nanoparticles[END_REF][START_REF] Florescu | Thermal emission and absorption of radiation in finite inverted opal photonic crystals[END_REF][START_REF] Biswas | Theory of subwavelength hole arrays coupled with photonic crystals for extraordinary thermal emission[END_REF][START_REF] Seager | Accurate measurements of thermal radiation from a tungsten photonic lattice[END_REF][32].

A key feature of thermal emission assisted by surface waves is that it is p-polarized in the plane perpendicular to the lines of the grating. Few studies have dealt with the polarization of the thermal emission. Yet, it has been shown that by modifying the local orientation of the grating for instance, it is possible to control the polarization of the emitted wave [START_REF] Dahan | Space-variant polarization manipulation of a thermal emission by a SiO2 subwavelength grating supporting surface phonon-polaritons[END_REF][START_REF] Dahan | Thermal image encryption obtained with a SiO2 space-variant subwavelength grating supporting surface phonon-polaritons[END_REF]. More recently [START_REF] Marquier | Anisotropic polarized emission of a doped silicon lamellar grating[END_REF], we have shown that surface wave assisted light emission could be also s-polarized out of the plane perpendicular to the grooves. In this paper we focus on the SiC sources of Ref. [START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF][START_REF] Greffet | Coherent emission of light by thermal sources[END_REF] to analyze the polarization of the thermal emission emitted in any direction. In particular, we report a study of the degree of polarization.

The SiC lamellar gratings are characterized by their period Λ, filling factor F and height h. These parameters are optimized to obtain 100% emissivity for a given wavelength. Two kinds of thermal light sources are considered : a quasi-isotropic source at a given wavelength or a directional thermal source. For both sources, we analyse the emission in planes which are not perpendicular to the grating lines. Notations for angles are summarized in Fig. 1.

Surface-plasmon excitation in s-polarization

In this section we show how surface waves can be used to emit s-polarized radiation in free space. It has been already observed in a paper by Inagaki et al. [START_REF] Inagaki | Plasma resonance absorptionin conical diffraction: effects of groove depth[END_REF] that reflectivity of a metallic grating can drop to zero when illuminated either in p or s-polarization in a direction which is not perpendicular to the the lines of a grating. It has been shown that surface plasmons are involved in the Figure 1: Emission of a grating in the direction (θ, φ). The electric field E makes an angle Ψ with the plane (z, k). Ψ = 0 • corresponds to the p-polarization and Ψ = 90 • to the s-polarization. absorption of s-polarized fields. There may be also polarization conversion upon reflection on a grating when exciting resonantly a surface plasmon [START_REF] Elston | Polarization conversion from diffraction gratings[END_REF]. This process can be viewed as the excitation of a surface plasmon by an p-polarized wave followed by the radiation of an s-polarized wave. All these processes clearly suggest that gratings supporting surface plasmons can radiate s-polarized light. The first thermal source that we study in this paper is a directional thermal source. The grating is characterized by its period Λ = 6.25µm, filling factor F = 0.5 and thickness h = 288nm. One can show in Fig. 2 the emissivity pattern of such a grating calculated for both p-and s-polarization. The conditions of coupling between surface waves and propagating waves have already been discussed in a previous paper [START_REF] Marquier | Anisotropic polarized emission of a doped silicon lamellar grating[END_REF]. According to this reference, there are two requirements in order to achieve an efficient coupling. The first one is obviously a phase matching condition, which is summarized in Fig. 3(a). This is the origin of the circle-like pattern that we can observe on Fig. 2 for both s and p-polarization. The second condition deals with the polarization of the emitted wave. Actually, the emitted wave must have an electric field component along the propagation direction of the surface wave. This is the origin of emissivity variations along the circles defined by the phase matching condition.To be more precise, a surface wave propagating along the interface with a wave vector k SP has two components of the electric field : one is parallel to k SP and the other is parallel to the normal of the surface. It follows that the surface current density associated with the surface plasmon in the material has a component parallel to k SP . An external electric field can excite a surface plasmon provided that it has a component parallel to k SP . This condition explains the φ dependence and the polarization dependence of the emissivity. Let us consider for example the absorption of a plane wave with an angle φ = 0 so that the incident wave vector is along the x-axis (k inc x , 0). A plasmon can in principle be excited with a wavector (k x = k SP , k y = 0) provided that k inc x = k SP -2π/Λ. If the incident plane wave is p-polarized, the incident field can excite a current density parallel to the x-axis and therefore excite the surface wave. Conversely, if the wave is s-polarized, the incident electric field is along the y-axis and therefore cannot excite a current density along the x-axis.

u x = k x / k u y = k y / k (a) u x = k x / k u y = k y / k (b)
We note that the coupling in s-polarization is not efficient so that the corresponding emissivity is very low, below 10%. This is not due to fundamental reasons but instead to the fact that this grating had been optimized to emit in p-polarization [START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF][START_REF] Greffet | Coherent emission of light by thermal sources[END_REF]. These low values of emissivity do not allow us to perform accurate measurements in s-polarization on such a source. We will see in the next section that an isotropic source instead of a directional one is more interesting to show both numerically and experimentally surface-wave assisted emission in s-polarization.

Isotropic source

The second thermal source is a quasi-isotropic source for a given wavelength. The grating is characterized by its period Λ = 3.0µm, filling factor F = 0.4 and thickness h = 350nm. The emission wavelength is λ = 10.88 µm and corresponds roughly to the asymptotic part of the dispersion relation of surface phonon polaritons, so that the emissivity is expected to reach one for this wavelength and to be very low for other wavelengths [START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF]. This can be viewed as a first step to the fabrication of efficient light sources which could emit monochromatic radiation in a very isotropic way. As in the case of directional sources, previous studies dealt only with the emission in a plane perpendicular to the grating lines. Figure 4 displays the emissivity pattern in both p-and s-polarization. One can see that the phase matching condition is always fulfilled (Fig. 3(b)). Figure 3: Dispersion relation of surface plasmon-polaritons in a (k x , k y ) plane for two gratings with different period at a fixed frequency. In both figures, the radius of the dashed circle is the wave vector modulus in free space. Inside this circle (in gray), the emitted waves have a real z-component of k, so that they are propagating waves. Outside the dashed circle, the waves are evanescent waves. In (a), the solid circle at the center has a radius equal to the modulus of the wave vector of the surface wave k SP . With a grating of period Λ, this solid circle is reproduced with a period 2π/Λ: it represents the surface waves diffraction at the orders ±1. One can see in the first figure (a), corresponding to the case of section 2, that a part of the solid line is lying now in the light cone, so that the corresponding surface polaritons are coupled to propagating waves. In the second figure (b), corresponding to the case of section 3, the period Λ is lower so that the linear part of the dispersion relation is always outside the light cone. Only the surface waves lying on the asymptotic part of the dispersion relation can be coupled to propagating waves. This is represented by the parallel red lines which cover the whole gray area. This means that propagating waves can be coupled to surface waves in every directions (k x , k y ) at the asymptotic wavelength.

The pattern given in Fig. 4 is due to the condition on the polarization of the emitted wave that we discussed above. By the way, in this case the emissivity in s-polarization goes to one in the direction parallel to the lines of the grating. In this direction, the emission is also isotropic. Observation of Figure 4 suggests that the emissivity pattern in p-and spolarization are complementary. It is possible to calculate an average emissivity for this grating. The result is shown in Fig. 5(a). The emissivity reaches noticeably the same value of about 0.5 in every directions. This value is sufficiently large to be experimentally detected. Thus, we measured the emissivity of this source. The grating was heated at about 500 • C. The experimental setup to measure and normalize the emitted flux to obtain emissivity is described in detail in ref. [START_REF] Marquier | Coherent spontaneous emission of light by thermal sources[END_REF]. The measurement is shown in Fig. 5(b) for three different directions given by θ = 20 • and φ = 00 • , 45 • and 90 • . We plotted the average emissivity in both p-and s-polarization. One can clearly see that the emissivity spectrum is the same for the three directions. It is important here to note that this effect is possible only at the wavelength corresponding to the asymtotic part of the dispersion relation of the surface wave. For this SiC grating, it corresponds to 11 µm. For other wavelengths, the grating behaves almost like the plane interface, that is to say that between 10.5 and 12.5 µm, the emissivity is very low, near 0.1. This grating allows the emissivity to reach 0.5 instead of 0.1 at λ = 11 µm.

u x = k x / k u y = k y / k (a) u x = k x / k u y = k y / k (b)

Dependence on the polarization

A light source is characterized not only by its intensity but also by its polarization. We have seen that the isotropic source emits light whose polarization depends on the emission angle. In order to fully determine the polarization characteristics, one has to derive the Stokes parameters. We shall use them to derive the degree of polarization of the emitted field [START_REF] Mandel | Optical coherence and quantum optics[END_REF]. The four Stokes parameters are obtained considering different polarizations of the emitted radiation: on the one hand, s-and p-polarization and on the other hand two linear polarizations corresponding to ψ = +45 • and -45 • , respectively called a-and b-polarization here, and two circular polarizations, left-handed and right-handed, respectively called L-and R-polarization. Stokes's parameters are defined by the combination of emitted intensities in these polarizations. For a thermal source, the Stokes parameters are linearly related to the polarized directional emissivity. As a result we have:

S 0 = ǫ s + ǫ p S 1 = ǫ s -ǫ p S 2 = ǫ a -ǫ b S 3 = ǫ L -ǫ R
It has been pointed out in two recent papers [START_REF] Tervo | Degree of coherence for electromagnetic fields[END_REF][START_REF] Setala | Stokes parameters and polarization contrasts in Young's interference experiment[END_REF] that a field whose intensity is uniform in a plane can have an interference structure. Indeed, the interference pattern produced by Young's experiments on a screen on which intensity (that is to say S 0 /2) is uniform can present a maximum contrast for other Stokes's parameters. In an other way, Dahan et al. made some studies on the manipulation of polarization by SiO 2 gratings supporting surface phononpolaritons [START_REF] Dahan | Space-variant polarization manipulation of a thermal emission by a SiO2 subwavelength grating supporting surface phonon-polaritons[END_REF]. They could use it for instance to encrypt thermal images [START_REF] Dahan | Thermal image encryption obtained with a SiO2 space-variant subwavelength grating supporting surface phonon-polaritons[END_REF]. Therefore, it appears that a polarization study of the light emitted by a SiC grating is important to fully characterize the thermal source. Figure 6 shows that the degree of polarization of this source strongly depends on the emission direction. In directions characterized by φ = 00 • , that is to say in the plane perpendicular to the lines of the grating, P is close to one. That means that the emitted radiation is fully polarized in this plane. On the contrary, P drops to zero when θ tends to 90 • in the plane parallel to the lines of the grating. Thus the beam is completely unpolarized. Actually, in the plane perpendicular to the lines, the only polarization which can be couple to a surface wave is the p-polarization. Consequently, the radiation is emitted in a polarization defined by a basis vector: this explains the large value of the degree of polarization in this plane. In other directions, both p-and s-polarized waves can be coupled to surface waves, it is thus possible to have the same emissivity for both polarizations and therefore a zero-degree of polarization.

u x = k x / k u y = k y / k

Conclusion

The polarization of thermal emission of SiC gratings has been studied. In particular, we focused on the emission in planes which are not perpendicular to the grooves of the gratings. It has been shown both numerically and experimentally that it was possible to couple surface phonons to s-polarized propagating waves. We have extended the scope of this work to a general study of the emitted radiation of such sources: Stokes's vectors and degree of polarization. We report the first study of polarized emission of a 1D grating in any direction. We have found that the overall emissivity does not depend on the polarization while the polarized emissivity strongly depends on the emission angle. It follows that the degree of polarization can vary between 0 and 1. Let us stress that that even if this paper deals with surface phonon on SiC, our conclusions are more generally valid for surface waves and can be applied to surface plasmon and metallic gratings [START_REF] Laroche | Highly directional radiation generated by a tungsten thermal source[END_REF][START_REF] Marquier | Anisotropic polarized emission of a doped silicon lamellar grating[END_REF].

Figure 2 :

 2 Figure 2: Polar representation of the emissivity at λ = 11.36 µm for the directional source in both p-polarization (a) and s-polarization (b) (numerical simulations).

Figure 4 :

 4 Figure 4: Polar representation of the emissivity at λ = 10.88 µm for the quasiisotropic source in both p-polarization (a) and s-polarization (b) (numerical simulations).

Figure 5 :

 5 Figure 5: Polar representation of the average emissivity in both p-and spolarization at λ = 10.88 µm for the quasi-isotropic source (a) (numerical simulation). Average emissivity of the same source measured with three different azimuthal angles φ at θ = 20 • (b) (experimental measurements).

Figure 7 :

 7 Figure 7: Polar representation of the degree of polarization P given by Stokes's parameters.

the emissivity for the a-, b-, L-and R-polarization. We obtain these results by computing the absorptivity of polarized plane waves. It must be noticed that the average of the emissivity either in both a-and b-polarization or in Land R-polarization gives the same result as in Fig. 5(a). This is not surprising considering the fact that we have in each case a basis of the polarization states. We can also observe that the emissivity patterns are very different from each other. The degree of polarization is defined from Stokes's parameters [START_REF] Mandel | Optical coherence and quantum optics[END_REF]:

Using Fig. 4 and Fig. 6, it is possible to calculate the degree of polarization of our thermal isotropic source. The result is shown in Fig. 7. Firstly, one can see


Maxime Rousselot
email: maxime.rousselot@harmonicinc.com

Éric Auffret

Xavier Ducloux

Olivier Le Meur
email: olivier.lemeur@irisa.fr

Rémi Cozot
email: remi.cozot@irisa.fr

Impacts of viewing conditions on HDR-VDP2

Keywords: Image quality, Image database, Dynamic range

HDR (High Dynamic Range) and WCG (Wide Color Gamut) increase significantly quality of viewing experience by rendering impressive images and videos. Automatic assessing the quality of these HDR WCG images is one crucial objective in broadcast process. Full-reference HDR metrics have been designed in the last years to achieve this objective:HDR-VDP2, HDR-VQM, PU-encoding metrics. Recent studies have pointed out that HDR-VDP2 is one of the best metric. Unfortunately, HDR-VDP2 is quite complex to use due to numerous and sometimes hard-to-know parameters such as display emission spectrum, surround luminance and angular resolution. In this paper, we show that HDR-VDP2 does not require an accurate knowledge of the viewing condition parameters. For that, we not only test the impact of these parameters on existing image databases of subjective quality scores, but also we propose a new and complementary image database made with a different HDR display.

I. INTRODUCTION

HDR (High Dynamic Range) and WCG (Wide Color Gamut) images/videos became in the last years an active field of research and standardization. HDR displays are capable of preserving more details in the bright and dark area of an image by increasing the peak brightness and lowering the black point brightness displayable. Some of these displays are also WCG, meaning that the available gamut is significantly bigger, allowing to display more saturated color. Assessing the quality of HDR and WCG content is a huge challenge. New metrics have been designed to this end. We can mention the HDR-VQM metrics [START_REF] Narwaria | HDR-VQM: An objective quality measure for high dynamic range video[END_REF] dedicated to HDR video and HDR-VDP2 [START_REF] Mantiuk | Hdr-vdp-2: a calibrated visual metric for visibility and quality predictions in all luminance conditions[END_REF] for still images. Some existing metrics used to assess the quality of SDR (Standard Dynamic Range) images have been also adapted to HDR by using a new encoding method called PU (Perceptual Unit) [START_REF] Aydın | Extending quality metrics to full luminance range images[END_REF].

These quality metrics are crucial in order to ensure a good quality of broadcasting. Previous studies suggested that the full-reference HDR-VDP2 metric is one of the best existing metrics for compression artifact. For example, in [START_REF] Hanhart | Benchmarking of objective quality metrics for hdr image quality assessment[END_REF], the authors assessed the performance of 35 quality metrics over 240 images compressed with JPEG XT [START_REF] Richter | On the standardization of the jpeg xt image compression[END_REF]. They concluded that HDR-VDP2 (version 2.2.1 [START_REF] Narwaria | HDR-VDP-2.2: a calibrated method for objective quality prediction of highdynamic range and standard images[END_REF]) and HDR-VQM metrics [START_REF] Narwaria | HDR-VQM: An objective quality measure for high dynamic range video[END_REF] were the best performing metrics. In [START_REF] Hanhart | Towards high dynamic range extensions of hevc: subjective evaluation of potential coding technologies[END_REF], authors came to the conclusion that HDR-VDP2 (but in an earlier version 2.1.1) can be successfully used for predicting the quality of video pair comparison contrary to HDR-VQM. More recently, Zerman et al. [START_REF] Zerman | An extensive performance evaluation of full-reference hdr image quality metrics[END_REF], first, combined several existing image databases and, second, they found out that HDR-VQM is the best full-reference HDR quality metric, closely followed by the HDR-VDP2.2.1 metric which gives similar results when one particular database is discarded.

Although that HDR-VDP2 is often considered as one of the best HDR metric, it is also important to point out that, in some cases, simpler full-reference metrics perform as well as HDR-VDP2. For instance, in [START_REF] Hanhart | Subjective and objective evaluation of hdr video coding technologies[END_REF], authors showed that HDR-VQM, PU-encoding VIF [START_REF] Sheikh | Image information and visual quality[END_REF] and PU-SSIM provide similar performances as HDR-VDP2. In [START_REF] Vigier | Performance and robustness of hdr objective quality metrics in the context of recent compression scenarios[END_REF], results indicate that PU-VIF and HDR-VDP2 have similar performances, although PU-VIF have a slightly better reliability than HDR-VDP2 for lower scores. Considering that HDR-VDP2 has similar results than much simpler metrics in some specific contexts, on one hand, and on the other hand, the complexity to use HDR-VDP2 which requires a number of parameters, we may ask whether it is worth to use this metric.

In this paper, we focus on HDR-VDP2 (version 2.2.1) metric. This metric is a bio-inspired metric that models accurately the early stages of the human vision such as intra-ocular scattering, spectral sensitivity of human eye photo-receptor and luminance masking. In addition, HDR-VDP2 requires several input parameters in order to define the visual environment in which image is seen. These important parameters are the surround luminance, the display spectral emission and the angular resolution. We investigate the sensibility to input parameters and the limitations of HDR-VDP2 by analyzing its performance both on existing HDR database image annotated with quality score and on a new and complementary database. The contributions of this paper are twofold:

• determining the influence of input parameters on the performance of HDR-VDP2; • providing a new and complementary database for evaluating the quality of existing HDR metrics. The paper is organized as follows. In Section II, we present existing image databases that are classically used for assessing the performance of HDR quality metric. We also introduce our new database. Section III presents a new analysis of HDR-VDP2, revealing that HDR-VDP2 is not sensitive to a number of input parameters. The last section concludes the paper.

II. DATABASES PRESENTATION A. Existing databases

For this study, three HDR image databases annotated with quality scores are used. Table I presents the main characteris-tics of these databases; a short description is also given below:

• Narwaria et al. [START_REF] Narwaria | Tone mapping-based high-dynamic-range image compression: study of optimization criterion and perceptual quality[END_REF] 1 's database is composed of 27 source images, which have been distorted by a backward compatible scheme, meaning that the HDR is first tonemapped to a standard dynamic range (SDR) and compressed using a standard JPEG codec. Images are then tone-expanded before being displayed on screen. HDR-VDP2 is the best performing metric for this database. • Korshunov et al. [START_REF] Korshunov | Subjective quality assessment database of HDR images compressed with JPEG XT[END_REF] 2 's database consists in images distorted using the backward-compatible JPEG-XT standard. HDR-VDP2 is also the best quality metric for this database, although without significant difference with HDR-VQM [START_REF] Hanhart | Benchmarking of objective quality metrics for hdr image quality assessment[END_REF]. • Zerman et al. [START_REF] Zerman | An extensive performance evaluation of full-reference hdr image quality metrics[END_REF] 3 's database is partially composed of images from [START_REF] Valenzise | Performance evaluation of objective quality metrics for hdr image compression[END_REF]. The distorted images are generated by using both backward-compatible distortion with codec JPEG, JPEG2000 and using a non backward-compatible scheme with the use of the Perceptual Quantizer (PQ) for the electro-optical transfer function (EOTF) first normalized in the SMPTE ST 2084 [START_REF]High dynamic range electro-optical transfer function of mastering reference displays[END_REF]. These three subjective experiments have been carried on using a SIM2 HDR47ES4MB display. This display is able to render values between 0.03 and 4000 cd/m 2 .

As mentioned earlier, the most used HDR databases have been built with the same SIM2 HDR47ES4MB display. We believe that this is a first limitation of existing databases. Moreover, the gamut used to encode the images is always the legacy gamut, although a lot of HDR images and videos, even BT.709 [START_REF]Parameter values for the HDTV standard for production and international programme exchange[END_REF] content, will be encapsulated in a wider gamut like BT.2020 [START_REF]Parameter values for ultra-high definition television systems for production and international programme exchange[END_REF]. Finally, as distortions included in current databases are mainly luminance-based distortions, it would be helpful to put more emphasis on color degradations. To overcome the aforementioned limitations, we propose a new database by using an other display, new kinds of distortions and BT.2020 encapsulation format.

B. A new database

1) Content creation: Eight images were selected from 3 collections: two are from the MPEG HDR sequences (FireEater and Market) [START_REF] Lasserre | Description of hdr sequences proposed by technicolor[END_REF], one is from the Stuttgart HDR Video Database [START_REF] Froehlich | Creating cinematic wide gamut hdr-video for the evaluation of tone mapping operators and hdr-displays[END_REF] and the remaining five images comes from HDR photographic survey [START_REF] Fairchild | The hdr photographic survey[END_REF]. Note that these images also belong to Zerman et al.'s database [START_REF] Zerman | An extensive performance evaluation of full-reference hdr image quality metrics[END_REF]. All these images have been encapsulated in the WCG gamut BT.2020 [START_REF]Parameter values for ultra-high definition television systems for production and international programme exchange[END_REF] instead of the standard gamut BT.709 [START_REF]Parameter values for the HDTV standard for production and international programme exchange[END_REF].

Four kinds of distortions have been chosen:

• HEVC compression using the recommendation ITU-T H Suppl.15 [START_REF]Conversion and coding practices for HDR/WCG Y'CbCr 4:2:0 video with PQ transfer characteristics[END_REF]. This means a PQ encoding [START_REF]Image parameter values for high dynamic range television for use[END_REF] and a 4:2:0 chroma sub-sampling using a luma-adjustment process. Four different quantizers (Qp) were selected for each image. The chroma Qp offset algorithm is used to overcome a compression issue with the chroma This is even more true for BT.709 content encapsulated in a BT.2020 gamut. This kind of encapsulation creates color artifact when the chroma components do not have enough bits allocated to it. • HEVC compression without the chroma Qp offset algorithm as described above. Three Qp were selected for each image. • Gaussian noise on the chroma components: 3 levels of noise were selected. • Gamut mismatch: two degradations were created for this category. On one hand, the BT.709 images were considered as if they had been already encapsulated in a BT.2020 gamut creating more saturated images. On the other hand, we took images already encapsulated in a BT.2020 gamut and considered them as BT.709 images and re-encapsulated them in a BT.2020 gamut. This creates less saturated images. 2) Protocol: Participants scored the quality of images by using the Double-Stimulus Impairment Scale (DSIS) variant I methodology [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF] with a side-by-side comparison. A continuous scale form 0 to 100 was used: 100 meaning imperceptible, 75 perceptible but not annoying, 50 slightly annoying, 25 annoying and 0 very annoying. Because of the side-by-side comparison, the images were cropped to 944 × 1080 with 32 black pixels in the middle.

Fifteen naive subjects participated in this test (11 male, 4 female) with an average age of 25.8. All declared normal or corrected-to-normal vision. One participant was removed from the analysis using the methodology described in [START_REF]Methodology for the subjective assessment of the quality of television pictures[END_REF].

Image pairs were displayed on Sony BVM-X300 monitor with a peak brightness measured at 1000 cd/m 2 . The luminance of a black pixel was too low to be measured by our equipment. The distance between the participant and the screen was 3.2 times the picture height, leading to an angular resolution of approximately 60 pixels per degree.

Ninety-six pairs of images were presented to the viewers, one side being always the reference. 50% of the participant had the reference always on the right-hand side, 50% always on the left-hand side. To avoid a bias with the order of presentation, the pairs of images were randomized for each participant with the condition that the same content was never shown consecutively. Each image pair was shown 10 seconds and voting time was 5 seconds. The test session lasts 35 minutes (including instructions and training time) with a 5 minutes pause in the middle of the test.

The distribution of the collected subjective scores, illustrated on Fig. 1, is almost uniform, indicating an appropriate choice of the different levels of distortion.

3) Objective metric: To simulate our display, we cropped the luminance range of the images between 0 and 1000 cd/m 2 .

The performances were evaluated using four metrics: the Pearson Correlation Coefficient (PCC), The Spearman Rank Order Coefficient (SROCC), the Outlier Ratio (OR) and The Root Mean Square Error (RMSE). These performance metrics were calculated after mapping the objective on the subjective score. This is done by applying a non-linear regression on the objective metric scores using a logistic function describe by Equation 1.

Y = a + b 1 + e -(cX+d) ( 1
)
where Y is the objective score map on the subjective score, X the objective score and a, b, c and d the parameters determined by the regression.

Table II gives the performance of HDR-VDP2, HDR-VQM, PU-VIF, PU-MSSIM on the proposed database. HDR-VDP2 and PU-MSSIM performs the best even if these performances are not significantly better than HDR-VQM (for all metrics except for RMSE). The significance was estimated using the methods described in [START_REF]Methods, metrics and procedures for statistical evaluation, qualification and comparison of objective quality prediction models[END_REF]. Both of them have a higher correlation than PU-VIF, PU-SSIM and PU-PSNR.

III. COMPREHENSIVE ANALYSIS OF HDR-VDP2

In this section, we elaborate on the performance of HDR-VDP2 on the proposed database. Then, we evaluate the influence of the user-defined parameters on the ability of the metric to predict quality scores.

A. Detailed analysis of HDR-VDP2 on the proposed database

In the previous section II-B3, Table II gives the overall performances of HDR-VDP2 on the proposed database. In order to go further in the analysis, we first plot the MOS in function of the corrected HDR-VDP2 scores as illustrated by Figure 2. Several observations can be made. First, the HDR-VDP2 metric behaves quite well for the different kinds of degradations. When considering the compression artifact only (blue and red dots in Figure 2), this metric performs well. Table III presents the performance of HDR-VDP2 as well as HDR-VQM and PU-VIF for the proposed artifact compression. We observe that PU-VIF and HDR-VDP2 have similar performances. HDR-VQM is the worst performing metric. The difference is significantly lower in term of OR with PU-VIF and in term of RMSE with both HDR-VDP2 and PU-VIF. We also observe that the MOS scores obtained for compressed images with and without Qp offset method are strongly correlated (pcc=0.95). This suggests that observers are consistent in their quality scoring whether or not the Qp offset method is used. One exception concerns the RedwoodSunset image, for which the use of Qp offset brings a significant add-value, as illustrated by Figure 3. In this case, HDR-VDP2 metric provides exactly the same score.

Despite the fact that HDR-VDP2 is based on the luminance channel only, HDR-VDP2 turns out to be quite resilient to chromatic distortions, except for the default gamut mismatch. This default was reported quite hard to evaluate by naïve observers. A rather strong color distortion that does not change the global aesthetic of the scenes can be assessed positively by observers.

B. Simplifying the usage of HDR-VDP2

1) Sensitivity to the screen spectral emission.: In this section, we will discuss about the sensitivity of HDR-VDP2 to the spectral emission of the display. For this purpose, we measured the spectral emission of 5 HDR displays, two Led LCD displays (Sony KD-75X9405C and SIM2 HDR47ES4MB) and three OLED displays (LG OLED 65 E6V,Sony BVM-X300 and Loewe Bild 7.55). Three patches of pure blue, green and Red were measured on displays using the probe X-rite Eye One Pro 2. We then measured a white patch to assess the spectral additivity of the components. The measured spectrums are plotted in Figure 4.

We then tested the performance of HDR-VDP2 when changing this parameter with this five spectrum and with a white spectrum (so we don't take into account the display characteristic), with a Dirac spectrum where the value of the Diracs correspond to the primary wavelengths define by the CIE and with a spectrum with only the blue component.

Results are reported in Table IV for Zerman et al. and the proposed database. Surprisingly, the spectrum modification has almost no effect on the performance metrics. Note that we observed a similar result for Narwaria et al. and Korsunov et al.'s databases. The variations of all the values aren't significant for all real spectrum. On the proposed database, the performance slightly decreases when using a white spectrum. When the spectrum is reduced to the blue component, the performance drops for the proposed database but stays stable for Zerman et al.'s database. Overall, this test suggests that only a coarse estimation of the display spectrum is required for evaluating the quality score with HDR-VDP2.

2) Sensitivity to the surround luminance: In this section, We measure the performance when using different values of the surround luminance. The PCC and SROCC are reported in Figure 5 (Top). OR and RMSE show similar trends. The surround luminance has a very small impact on HDR-VDP2 performance, which slightly decreases for low luminance for our database and Narwaria et al..

3) Sensitivity to the angular resolution: As in the previous section, we estimate the sensibility of HDR-VDP2 to the angular resolution. The PCC and the SROCC are given in Figure 5 (Bottom). Performances are stable in the range 30 to 80 pixel/degree. Below 30 pixel/degree performances steadily drop down. It might be due to the pooling phase of HDR-VDP2, in which the weights associated with the lowest frequency sub-bands are almost null (when the angular resolution diminish, the viewing distance decreases and the amount of low-frequencies increases).

However, it is important to emphasize that, if the values of the metric did not significantly change in previous sections, changing the angular resolution parameter shifts HDR-VDP2 scores without losing in correlation. For example, the mean value of HDR-VDP2 scores on our database is 60.3 when the parameter is 30 pixel/degree and 48.9 when the parameter is equal to 60 pixel/degree.

IV. CONCLUSION

In this paper, we investigated the sensibility of HDR-VDP2 metrics to three user-defined parameters, namely the angular resolution, the spectrum emission and the surround luminance. Our results suggest that only an approximate estimation of the viewing conditions is required. Indeed, in the context of this study, HDR-VDP2 quality scores are not sensitive to moderate variations of viewing condition parameters. It should be noted however that this is valid only for the quality scores and not for the distortion map.

To go deeper into the analysis, we proposed a new database, including new kinds of distortion. We found out that HDR-VDP2 has a good resilience to chromatic artifact, although this is a luminance-only metric (a similar comment can be made for simpler metrics such as PU-MSSSIM). This observation raises some questions about HDR images and their quality evaluation. As HDR images are still quite confidential, we may ask the following question: are naïve observers able to evaluate small to medium distortions on this new format? We believe that more experienced viewers could have been more sensible to the color distortions, which are much more complex to assess than luminance artifacts.

Fig. 1 :

 1 Fig. 1: MOS Distribution for the proposed database.

Fig. 2 :

 2 Fig. 2: HDR-VDP2 scores in function of the MOS.

Fig. 3 :

 3 Fig. 3: Subjective and objective scores in function of HEVC Qp for the RedwoodSunset image.

Fig. 4 :

 4 Fig. 4: Spectral emission of displays

TABLE I :

 I Existing HDR databases and the proposed one.

	Name	#Obs	#Img	Protocol	Distortion	Display	Gamut	Surround	Angular
								luminance	resolution
	Narwaria et al. [16]	27	140	ACR-HR	JPEG	SIM2 HDR	BT.709	130cd/m 2	60 pix/deg
	Korshunov et al. [13]	24	240	DSIS	JPEG-XT	SIM2 HDR	BT.709	20cd/m 2	60 pix/deg
				(side by side)
	Zerman et al. [24]	15	100	DSIS	JPEG, JPEG-XT	SIM2 HDR	BT.709	20cd/m 2	40 pix/deg
				(side by side)	JPEG2000
	Proposed one	15	96	DSIS	HEVC, Gaussian noise	Sony BVM-X300	BT.2020	40cd/m 2	60 pix/deg
				(side by side)	Gamut mismatch

TABLE II :

 II Performances on the proposed database.

	Metric	PCC	SROCC	OR	RMSE
	HDR-VDP2	0.89	0.87	0.48	12.5
	HDR-VQM	0.85	0.83	0.56	14.7
	PU-VIF	0.77	0.75	0.58	17.7
	PU-MSSSIM	0.89	0.87	0.46	12.2
	PU-SSIM	0.71	0.70	0.59	19.7
	PU-PSNR	0.80	0.78	0.52	16.7

TABLE III :

 III HDR-VDP2 scores on compression degradations.

	Metric	PCC	SROCC	OR	RMSE
	HDR-VDP2	0.90	0.87	0.45	9.3
	HDR-VQM	0.85	0.83	0.61	11.45
	PU-VIF	0.90	0.88	0.36	9.25
	PU-MSSSIM	0.92	0.90	0.46	11.0
	PU-SSIM	0.65	0.63	0.75	21.7
	PU-PSNR	0.83	0.82	0.54	15.9

TABLE IV :

 IV Performance of HDR-VDP2 on Zerman et al. and on the proposed database.

	ZERMAN ET AL.
	Spectrum	PCC	SROCC	OR	RMSE
	SONY BVM-X300	0.94	0.93	0.45	10.4
	LG OLED 65 E6V	0.94	0.93	0.47	10.4
	Sony KD-75X9405C	0.93	0.93	0.47	10.4
	Loewe Bild 7.55	0.93	0.93	0.47	10.6
	SIM2 HDR47ES4MB	0.93	0.92	0.46	10.6
	white spectrum	0.93	0.91	0.55	11.8
	Diracs spectrum	0.91	0.92	0.50	11.0
	blue-only spectrum	0.88	0.87	0.55	13.9
	PROPOSED DATABASE
	Spectrum	PCC	SROCC	OR	RMSE
	SONY BVM-X300	0.89	0.87	0.48	12.5
	LG OLED 65 E6V	0.90	0.87	0.47	12.3
	Sony KD-75X9405C	0.90	0.87	0.46	12.4
	Loewe Bild 7.55	0.90	0.87	0.48	12.3
	SIM2 HDR47ES4MB	0.90	0.87	0.48	12.2
	white spectrum	0.84	0.87	0.59	14.9
	Diracs spectrum	0.90	0.87	0.46	12.4
	blue-only spectrum	0.67	0.67	0.68	20.5


Trung-Hien Nguyen

Pascal Scalart

Mathilde Gay

Laurent Bramerie

Christophe Peucheret

Fausto Gomez-Agis

Olivier Sentieys

Jean-Claude Simon

Michel Joindot

T.-H Nguyen

J.-C Simon

T.-H Nguyen

‡ P Scalart

§ F Gomez-Agis

New Metric for IQ Imbalance Compensation in Optical QPSK Coherent Systems

Keywords: Coherent communications, Fiber optical communications, IQ imbalance, Modulation

We report on a simple alternative method for the compensation of quadrature imbalance in optical quadrature phase-shift-keying (QPSK) coherent systems. By introducing a new metric, the phase imbalance can be determined and compensated. The proposed method is theoretically and numerically analyzed. In particular, it is shown that the method exhibits a small bias of estimated phase imbalance value. Thanks to its deterministic property, this bias can be simply compensated by incorporating at the receiver a phase rotator (or phase shift) whose value can be determined based on an analytical analysis. Moreover, the algorithm is also experimentally validated through bit-error-rate (BER) and error vector magnitude (EVM) measurements. A good agreement on the performance of the proposed method with that of the Gram-Schmidt orthogonalization procedure (GSOP) is shown in a 20 Gbit/s optical QPSK experiment. The robustness of both methods was verified with up to 30 • phase imbalance by comparing the signal with and without phase imbalance compensation. A 10% reduction of EVM is achieved with our method for a high phase imbalance of 30 • while the implementation complexity can be reduced owing to the suppression of the use of square-root operators.

Introduction

Optical transmission systems using polarization division-multiplexed quadrature phase-shiftkeying (PDM-QPSK) and coherent detection at 100 Gbit/s have been the object of intensive investigations and are already commercialized. One of the aims of the optical fiber communication industry is now to move towards systems carrying over 100 Gbit/s per wavelength channel [START_REF] Raybon | High symbol rate coherent optical transmission systems: 80 and 107 Gbaud[END_REF]. This trend towards always increasing capacities has activated a renewed interest in coherent optical communication systems thanks to their improved receiver sensitivity and spectral efficiency, and ability to mitigate transmission impairments in the digital domain [START_REF] Ip | Coherent detection in optical fiber systems[END_REF].

As in in-phase/quadrature (I/Q) down-conversion radio architectures [START_REF] Valkama | Advanced methods for I/Q imbalance compensation in communication receivers[END_REF], the I and Q components of the optical field should be ideally orthogonal to each other in an optical QPSK coherent system. However, hardware implementation imperfections and finite tolerances of the front-end components, such as incorrect bias-point settings in the modulator, imperfect splitting ratio of couplers, photodiodes responsivities mismatch and incorrect adjustment of polarization splitters can create amplitude and phase imbalance, known as quadrature imbalance (or IQ imbalance), which destroys the orthogonality between components of the received signal [START_REF] Liu | Receiver sensitivity improvement in optical DQPSK and DQPSK/ASK through data-aided multi-symbol phase estimation[END_REF].

Since digital signal processing (DSP) circuits are becoming increasingly faster, providing simple and efficient compensation of linear [START_REF] Savory | Digital filters for coherent optical receivers[END_REF] and, possibly, non-linear [START_REF] Ip | Compensation of dispersion and nonlinear impairments using digital back propagation[END_REF] impairments, it is important to assess their potential for the compensation of this detrimental loss of orthogonality in the receiver. Some effort has been dedicated to the compensation of IQ imbalance with the help of DSP. IQ imbalance causing inter-carrier interference (ICI) due to imperfect image rejection in multicarrier direct-detection systems has been analytically and numerically investigated [START_REF] Valkama | Advanced methods for I/Q imbalance compensation in communication receivers[END_REF], and extended to deal with frequency-dependent IQ imbalance [START_REF] Anttila | Circularity-based I/Q imbalance compensation in wideband direct-conversion receivers[END_REF]. In multicarrier coherent-detection systems, several frequency domain compensation methods for IQ imbalance have been reported, both for wireless and optical transmission. Those include joint transmitter (Tx) and receiver (Rx) IQ imbalance compensation [START_REF] Schenk | Estimation and compensation of frequency selective TX/RX IQ imbalance in MIMO OFDM systems[END_REF]- [START_REF] Liu | Joint Tx and Rx IQ imbalance compensation of OFDM transceiver in mesh network[END_REF] for the blind compensation class (in the sense that no known training symbols are required), the pilot-assisted method [START_REF] Chung | Transmitter IQ mismatch compensation in coherent optical OFDM systems using pilot signals[END_REF], the use of a novel training symbol structure [START_REF] Ma | Novel training symbol structure for transmitter IQ mismatch compensation for coherent optical OFDM[END_REF], or the joint compensation of phase noise and IQ imbalance [START_REF] Cao | Decision-aided joint compensation of transmitter IQ mismatch and phase noise for coherent optical OFDM[END_REF]. For single carrier optical coherent detection systems, some work exploiting timedomain compensation has been numerically and experimentally carried out. More specifically, IQ imbalance has been corrected in optical coherent QPSK systems by applying different methods such as the Gram-Schmidt orthogonalization procedure (GSOP) [START_REF] Fatadin | Compensation of quadrature imbalance in an optical QPSK coherent receiver[END_REF], the ellipse correction method (EC) [START_REF] Chang | Impact of quadrature imbalance in optical coherent QPSK receiver[END_REF], and IQ compensation based on the constant modulus algorithm (CMA) [START_REF] Petrou | Quadrature imbalance compensation for PDM QPSK coherent optical systems[END_REF] or the statistical properties of received signals [START_REF] Qiao | Quadrature imbalance compensation algorithm based on statistical properties of signals in CO-QPSK system[END_REF].

In this paper, we comprehensively investigate an alternative method for IQ imbalance compensation [START_REF] Nguyen | IQ imbalance compensation based on maximum SNR estimation in coherent QPSK systems[END_REF] based on the definition and computation of a suitable novel metric for the detected signal, in order to estimate and compensate for the phase imbalance. The approach using this new metric, called best-matched signal estimation method (MSEM), provides an interesting alternative to existing algorithms thanks to its reduced complexity and a comparable performance. More particularly, our main contributions in this paper are: (i) a comprehensive analytical analysis of the proposed IQ imbalance compensation method; (ii) numerical validation of the derived analytical expression in the presence of additive noise; (iii) finding and numerically proving the existence of a small deterministic bias of the estimated phase imbalance. These new contributions strongly confirm our experimental results of the proposed IQ imbalance compensation method. Moreover, the proposed method requires no square-root operators, which can make its implementation in hardware platforms simpler. Furthermore, our method provides information on the phase imbalance value between I and Q components, which can be helpful for the characterization and calibration of the IQ modulator or the coherent receiver. The accuracy of our method is compared to that of the GSOP approach by evaluating the bit-error-rate (BER) and the error vector magnitude (EVM) of compensated constellations. The performance of the proposed MSEM method matches that of the GSOP, while its implementation complexity is reduced.

This paper is organized as follows: Section 2 presents in details an analytical analysis of the proposed method for IQ imbalance compensation based on the introduction of a new metric. Numerical simulations of the proposed method are carried out in Section 3 to demonstrate the effectiveness of the method. Section 4 describes the experimental setup used to validate this work, along with a discussion of the achieved results in Section 5. Finally, conclusions are drawn in Section 6.

Analytical analysis

In our study, we focus only on phase imbalance since amplitude imbalance can be corrected at the hardware level (for instance thanks to the use of automatic gain controlled trans-impedance amplifiers) [START_REF] Razavi | Design considerations for direct-conversion receivers[END_REF]. More specifically, the correction of phase imbalance introduced at the receiver is investigated with the assumption of perfect orthogonality between the I and Q components in the transmitted signals. In order to demonstrate how the maximization of a proposed metric can lead to IQ imbalance compensation, we first define and derive an analytical expression for such a new parameter for a QPSK signal. Without loss of generality, the QPSK signal at the transmitter is assumed to have unit power. φ S is the information carrying phase, which takes four possible values (2k + 1) • π/4, k = 0, 1, 2, 3 with the same probability. During transmission over an optical channel, this signal is corrupted by amplified spontaneous emission (ASE) noise, whose complex envelope n = n C + j • n S can be modeled as additive white Gaussian noise (AWGN). The QPSK signal is then detected by mixing with a local oscillator in a coherent receiver made from an optical 90 • hybrid and a pair of balanced photodiodes to extract the inphase (I) and quadrature (Q) components, as illustrated in Fig. 1. The resulting two baseband electrical components are obtained by the sum of quadratic detection in balanced photodiodes. To simplify our analysis, the carrier frequency offset (CFO) between the transmitter and receiver lasers is assumed to be zero, other noise sources (i.e. shot noise, thermal noise) are assumed to be negligible, and the power of the received samples is normalized to one. We can then express the I and Q terms by the following equations

I ∝ Re {E S E * LO } , (1)
Q ∝ Im E S E * LO .e jφ mis , (2)
where E S and E LO represent the fields of the signal (S) and local oscillator (LO) at the inputs of the 90 • hybrid in the receiver, Re and Im denote the real and imaginary parts of the beating products between the S and LO, respectively. φ mis represents the phase imbalance (or mismatch) stemming from phase misalignment between the two branches created in the coherent receiver. This term cancels for an ideal optical 90• hybrid. It is noted that imperfections in the implementation of the QPSK modulator could also lead to φ mis = 0 and consequently to a loss of orthogonality. However, the focus of the present analysis is on phase imbalance at the receiver and φ mis lies in the range [-90 • , 90 • ].

The I and Q components can be expressed by equations ( 1) and (2) as follows

I = cos φ S + n C Q = sin (φ S + φ mis ) + n C sin φ mis + n S cos φ mis , (3)
where n C and n S are zero mean random Gaussian variables with the same variance σ 2 that are furthermore assumed to be uncorrelated. The MSEM method consists in applying a phase shift, φ var , to the received signal,

R = I + j • Q, to form a new quantity Q n = Im R • e j•φvar
and consider it as a new quadrature component. A new metric, D, is introduced and defined according to

D = E 2 |r| 2 σ 2 |r| 2 , ( 4
)
where E(•) and |•| are the mean and modulus operators, respectively. σ 2 |r| 2 represents the variance of the squared modulus of the modified signal, r = I + j • Q n . This new metric is similar to the use of high-order statistics of circular signals to solve the source separation problem [START_REF] Cardoso | Equivariant adaptive source separation[END_REF]. Although higher (≥ 4th)-order statistics of the signal can be utilized [START_REF] Cardoso | Equivariant adaptive source separation[END_REF], only second-order statistics are used in our case to keep a reasonable complexity while providing an acceptable estimation accuracy.

After some mathematical manipulation, Q n can be written as

Q n = cos φ S • sin φ var + sin (φ S + φ mis ) • cos φ var + n C • sin φ var + (n C sin φ mis + n S cos φ mis ) • cos φ var . (5)
To simplify the calculations, the following quantities are defined

α = sin φ var + sin φ mis • cos φ var β = cos φ mis • cos φ var . (6)
The expected value of |r| 2 (calculated in Appendix 6) is derived as

E |r| 2 = 1 2 + σ 2 • 1 + α 2 + β 2 . ( 7
)
The variance of |r| 2 is given by (Appendix 6)

σ 2 |r| 2 = α 2 β 2 + 2 1 + 2α 2 + α 2 + β 2 2 • 1 + σ 2 • σ 2 . ( 8
)
Finally, by substituting ( 7) and ( 8) into (3), the following expression of D can be found

D = 1 2 + σ 2 2 • 1 + α 2 + β 2 2 α 2 β 2 + 2 1 + 2α 2 + (α 2 + β 2 ) 2 • (1 + σ 2 ) • σ 2 . ( 9
)
In the noiseless case (σ 2 = 0), introducing the expressions of α and β, given by ( 6), the proposed D metric without (wo) noise impact, D (wo) is given by An analysis of Eq. [START_REF] Chung | Transmitter IQ mismatch compensation in coherent optical OFDM systems using pilot signals[END_REF] shows that D (wo) versus φ var exhibits several maxima. In practical implementations, the IQ imbalance due to receiver imperfections should be sufficiently well controlled to remain in the range [-30 • , 30 • ]. In this range, D (wo) exhibits only one maximum corresponding to (sin φ var + sin φ mis • cos φ var ) = 0 or sin φ mis = -tan φ var . When the phase imbalance is small, φ var is close to -φ mis .

D (wo) = 1 + sin φ var • cos φ var • sin φ mis cos φ mis • cos φ var • (sin φ var + sin φ mis • cos φ var ) 2 . ( 10
Based on this observation, the MSEM method therefore relies on introducing a new parameter φ var in the received signal, and the value of φ var which maximizes D (Eq. ( 3)), will provide an estimator φmis of the opposite of the phase imbalance value, or in other words, φmis = -arg max φvar (D (φ var )). By correcting the received signal with the estimated value of the phase imbalance φmis , the IQ imbalance can be compensated for. The following paragraph will be devoted to the accuracy evaluation of the proposed algorithm.

Fig. 2 depicts the evolution of the estimated phase imbalance based on the MSEM method as a function of a predefined phase imbalance in the noiseless case. A slight deviation of the estimated phase imbalance compared to the actual one is observed, especially when the absolute value of the phase imbalance exceeds 15 • . Below this value, this estimation error is negligible. The estimated phase imbalance error is only 3.5 • for a phase imbalance of ±30 • . Note that, due to the deterministic nature of the bias introduced by the algorithm at high phase imbalance values, we can correct this issue by simply adding a specific value based on the analytical results. The proposed estimator and its bias are further studied in the presence of noise in the following section.

IQ compensation phase, ? var (degree)

Numerical simulation

In the presence of noise (σ 2 = 0), the derivation of an analytical expression for D is rather complex. Hence, the corresponding exact expression of the IQ compensation phase term that maximizes D is also complicated. Actually, if φ 0 is a phase value that maximizes D (wo) according to Eq. ( 10), it is shown in Appendix 6 that the first derivative of D(φ) at φ = φ 0 is not equal to 0 in the presence of noise. As a consequence, the maximum value of D and the corresponding estimated phase imbalance may be dependent on the SNR of the input received samples. For this reason, we numerically study the proposed method based on Eq. ( 9). To this aim, the phase imbalance, φ mis , is set to -15 • and assumed to be unknown at the receiver. Because the phase imbalance in optical coherent detection was assumed to be within the range [-30 • , 30 • ], the IQ compensation phase value, φ var , used to estimate the phase imbalance is also varied within this range. The amplitude noise is represented by its variance, σ 2 , in Eq. ( 9). Considering the unit power of the received signal, the signal SNR can be directly linked to σ 2 . As a consequence, we can calculate the evolution of D as a function of the IQ compensation phase, φ var , according to Eq. ( 9) for different SNR values, as shown in Fig. 3. Even though the noise level is varied, the evolution of D still exhibits one extremum in the considered range of [-30 • , 30 • ], similarly to the previous result in the absence of noise. Particularly, it is seen that the maximum values of D fall close to the opposite value of the preset phase imbalance of -15 • . Even in the presence of a high level of noise (SNR = 4 dB), the maximum value of D still occurs at an IQ compensation phase value, of about 15 • , which is close to the opposite value of the phase imbalance, -φ mis , confirming the proposed criterion for IQ phase imbalance estimation is still effective in the presence of high noise levels. It should be noted that the received samples SNR values are varied from 4 dB to 40 dB, in which the smallest SNR value is chosen corresponding to the theoretical BER of about 10 -2 of the QPSK signal [START_REF] Vitthaladevuni | A recursive algorithm for the exact BER computation of generalized hierarchical QAM constellations[END_REF], which corresponds to the forward-error coding (FEC) threshold assuming 28% overhead [START_REF] Mizuochi | Recent progress in forward error correction and its interplay with transmission impairments[END_REF].

The robustness of the proposed algorithm in the presence of noise is further investigated with different predefined phase imbalances, φ mis . Fig. 4 presents the estimated phase imbalance, φmis , as a function of the input received samples SNR for three predefined phase imbalance values of 10 • , 20 • and 30 • . It can be observed that the estimated phase imbalance exhibits an asymptotical limit as the input SNR increases. At 10 • phase imbalance, the impact of the input SNR on the proposed algorithm is negligible, resulting in no estimation error of the phase imbalance value. As the phase imbalance increases to 20 • , an input SNR of 16 dB is required to reach the asymptotical limit, whose value is about 1.1 • away from the pre-defined phase imbalance. This value agrees well with the theoretically calculated bias of the MSEM estimator at 20 • phase imbalance, in the absence of noise (Fig. 2). Note that, the difference between the estimated phase imbalance at low input SNR (i.e. 4 dB) and that at high input SNR (i.e. 20 dB) is small (about 1 • ). For a phase imbalance of 30 • (which is the maximum considered value), up to 20 dB input SNR is needed to achieve the asymptotical limit level. In this case, a 3.4 • difference between the asymptotical limit and the pre-defined phase imbalance matches the predicted bias in the theoretical calculation (Fig. 2). For this high phase imbalance, the predicted phase imbalance at the smallest considered SNR of 4 dB is about 2.5 • different from the asymptotical limit. The numerical calculation in the presence of noise confirms again the effectiveness of the proposed MSEM estimator. More specifically, the IQ compensation phase, φ var , resulting in the maximum value of D always happens around the opposite value of the predefined phase imbalance, φ mis , confirming the validity of the criterion used for phase imbalance estimation (i.e. φmis = -φ var in which φ var maximizes D). Even though this estimator exhibits an estimation error, thanks to the deterministic nature of the algorithm bias at high phase imbalance values, this issue can be corrected by simply adding a specific value based on the analytical results (as mentioned in the previous section). This correction can be perfect for high SNR values (> 16 dB), regardless of the phase imbalance value in the range [-30 • , 30 • ]. For low SNR values (≤ 16 dB), the correction is still valid with a negligible error, provided that the absolute phase imbalance value is less than 15 • . When the absolute value of phase imbalance is superior to 15 • , the bias correction introduces an estimation error smaller than 2.5 • . In practice, a lookup table would be used to apply a correcting value for each phase imbalance estimation.

The proposed method will be experimentally validated in the following section. The proposed algorithm is also compared to the Gram-Schmidt orthogonalization procedure with a 20 Gbaud QPSK signal.

Experimental setup

The impact of non-orthogonality between the I and Q components of an optical QPSK signal is then experimentally investigated using the coherent detection system shown in Fig. 5. To simplify the setup and avoid any impact from other functionalities implemented in the digital signal processing (DSP), only one polarization and back-to-back configuration are considered in this experiment. However, it is possible to extend the implementation to a polarization diversity receiver. The laser used at the transmitter side also acts as LO at the receiver side in order to ease the carrier frequency offset compensation. The estimated linewidth of this optical source tuned at a wavelength of 1540 nm is about 100 kHz. Two pseudo-random binary sequences (PRBSs) with lengths of 2 15 -1 and 2 23 -1 are used as inputs to an IQ modulator, resulting in a 10 Gbaud non return-to-zero (NRZ)QPSK signal at the modulator output. The bit sequences are synthesized using arbitrary waveform generators (AWG). The use of a variable optical attenuator (VOA) followed by an erbium-doped fiber amplifier (EDFA) and a 3 nm optical bandpass filter (OBPF) allows the adjustment of the optical signal-to-noise ratio (OSNR). At the receiver side, an optical 90 • hybrid (100 Gbit/s DP-QPSK integrated receiver) combines the QPSK signal and the LO. Two pairs of balanced photodiodes with a bandwidth of 32 GHz enable to retrieve the in-phase and quadrature components of the signal. Finally, the electrical signals samples are acquired by a real time oscilloscope with an electrical bandwidth of 16 GHz and sampling rate of 20 GS/s. The phase shift of the hybrid circuit in the coherent receiver cannot be varied away from 90 • . Based on the recommendations of the Optical Internetworking Forum (OIF) [START_REF]OIF: Implementation agreement for integrated dual polarization intradyne coherent receiver[END_REF], the absolute phase imbalance value of 90 • hybrid circuits should be less than 5 • . Such a value is supposed to have a negligible impact in our study. In order to investigate the proposed algorithm, we therefore introduce an adjustable phase imbalance at the IQ modulator by easily tuning the voltage applied to the phase-shifter controlling the relative phase between the two arms of the IQ modulator.

The post-processing is performed offline using the Matlab environment with the following steps: (i) IQ imbalance estimation and compensation with the investigated algorithms; (ii) equalization [START_REF] Proakis | Digital Communications[END_REF] to compensate for inter-symbol interference (ISI) induced by the limited bandwidth of equipments and decimation to the baud rate; (iii) laser phase noise estimation [START_REF] Viterbi | Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission[END_REF]; (iv) BER and EVM calculations. The algorithms for IQ imbalance compensation are applied to blocks of 10000 samples. Moreover, phase ambiguities are removed by processing the first 5000 samples to minimize the BER. Finally, the decoded bit sequence is compared to the transmitted one to determine the BER over 4 millions samples.

The popular EVM metric is utilized to evaluate the deviation of the reconstructed QPSK signal constellation compared to the reference one. As given in [START_REF] Mckinley | EVM calculation for broadband modulated signals[END_REF], the EVM can be expressed mathematically as

EV M = 1 N N k=1 |I ideal,k -I measure,k | 2 + |Q ideal,k -Q measure,k | 2 , ( 11
)
where N is total number of measured symbols. I ideal(measure),k and Q ideal(measure),k are the normalized voltages of ideal (measured) in-phase and quadrature components, respectively, for the k-th symbol. The achieved results will be discussed further in the following section.

Results and discussion

Fig. 6(a) shows the experimental constellation without IQ imbalance when the signal OSNR is 12 dB (all OSNR values are specified over a 0.1 nm noise bandwidth). Fig. 6(b) represents the constellation of the degraded signal with 15 • phase imbalance between the I and Q components. This signal degradation leads to detection errors, as it will be shown later. This degraded signal is processed with the same DSP procedure that deploys either the GSOP or the proposed MSEM method, resulting in the constellations in Fig. 6(c) and Fig. 6(d), respectively. The retrieved constellations are similar to the one without IQ imbalance, which illustrates the efficiency of the quadrature imbalance correction. In the following step, our method is experimentally validated by comparing its performance to that of the GSOP method in terms of BER and EVM. Fig. 7 depicts the measured BER as a function of received OSNR in different cases. The theoretical curve for the BER of a QPSK signal without IQ imbalance is based on the analysis in [START_REF] Proakis | Digital Communications[END_REF]. The experimental measurement without IQ imbalance fits the theoretical trace within experimental errors. Next, the phase imbalance is tuned resulting in a BER degradation and

GSOP 4N -3 * 6N + 4 * 2 MSEM (5N -1) • M † (4N + 5) • M † 0
* N -total number of samples used for IQ imbalance compensation † M -number of phase tests in MSEM method an OSNR penalty of 1.7 dB at a BER of 10 -3 . The phase imbalance is estimated at 17 • by the algorithm. The BER curves obtained after phase imbalance compensation using the GSOP and the MSEM methods are also presented. In both cases, the IQ imbalance is well compensated, reducing the power penalty close to the experimental case without IQ imbalance. Both methods are seen to result in equivalent BER improvement.

To further investigate the accuracy of the proposed algorithm, the EVM is calculated for different phase imbalance and different OSNR values. In each case φ mis is evaluated and the corresponding EVM is plotted in Fig. 8. Note that, the phase imbalance is estimated based on the output of the MSEM algorithm. It can be observed that, as the IQ imbalance increases, and when no compensation is applied, the EVM increases to 40% in the worst case, representing an important deviation from the optimal constellation. With both correction methods -MSEM and GSOP -and with an OSNR higher than 10 dB, the EVM remains almost constant for a phase imbalance lower than 17 • . Both methods can partially compensate for higher phase imbalances providing about 10% reduction of EVM value with the advantage of evaluating the phase imbalance in our method. Finally, for an OSNR of 8 dB, the EVM starts to increase when a phase imbalance value is superior to 9 • .

In term of hardware complexity, we qualitatively compare the required number of real adders, multipliers and square-root operators of the GSOP method with those of our proposed method, regardless of the complexity of the equalizer and other DSP blocks. It is assumed that the same total number of samples, N , is used for the different IQ imbalance compensation methods. M is the number of phase tests used in the MSEM method. Note that computing the modulus of a complex number requires 2 real multipliers and 1 real adder. Table 1 provides the summarized hardware complexities of the GSOP and MSEM methods for different operators. It can be observed that the required total number of multipliers and adders of the GSOP method is nearly M times less than that of the MSEM method due to the phase search iterations. However, the MSEM method does not require the square-root operator as for the GSOP method, bringing a potential simpler hardware implementation. It is worthy to note that the MSEM can be optimized to reduce the number of phase search iterations. In fact, based on the general equation of D (Eq. ( 9)), we can approximate the general form of the function D(φ var ) as a bell shape (as illustrated in Fig. 3). By using this approximation, the maximum D value and its corresponding φ var can be estimated with only 3 phase test values. The total number of adder and multiplier operators of the MSEM method would in this case be about 3-time larger than that of the GSOP method. A detailed optimization of the MSEM method is out of the scope of this paper.

As far as optical coherent m-QAM (i.e. 16-QAM) systems are concerned, the MSEM method may be adapted by cascading an extra step, called QPSK partitioning, as reported in [START_REF] Fatadin | Laser linewidth tolerance for 16-QAM coherent optical systems using QPSK partitioning[END_REF]. By applying this technique, we only need to consider the outermost selected constellation points, which are similar to a QPSK constellation, to estimate the IQ phase imbalance. However, the MSEM may require more samples than the GSOP to maintain comparable performance, owing to the use of only outermost constellation points. This extension is also outside the scope of this paper and its implementation is straightforward. Note that, the GSOP method is also applicable for m-QAM signal constellations, for instance in [START_REF] Winzer | Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM[END_REF]. In the transmission context where the impact of chromatic dispersion (CD) may impact the algorithms' performance, it is foreseen that the performance of our proposed algorithm will be similar to that of the GSOP, since both algorithms are based on the statistical properties of the received signal. A detailed study is left for future works.

Conclusion

We have proposed and demonstrated the MSEM method as an alternative solution for quadrature imbalance compensation in QPSK systems. An analytical form of a relevant new metric used to estimate quadrature imbalance has been derived. For phase imbalance superior to 15 • , the method presents an estimated phase imbalance deterministic bias which can be fully removed regardless of the value of the phase imbalance provided it is within the range [-30 • , 30 • ]. The accuracy of this method is also experimentally validated by comparing it with the GSOP method by means of BER and EVM measurements. The new compensation method provides a good correction with a potential reduced implementation complexity, even when the phase imbalance reaches values as high as 30 • .

Finally, the expected value of |r| 2 is derived by (as in Eq. ( 7))

E |r| 2 = 1 2 + σ 2 • 1 + α 2 + β 2 . (A.6)
The following part is the derivation of the variance of |r| 2 . The variance of |r| 2 can be written as

Var |r| 2 = E |r| 4 -E |r| 2 2 . (A.7)
Due to the assumption of central and uncorrelated variables, some terms cancel and the expectation of |r| 4 can be written as

E |r| 4 = E U 2 + V 2 2 + E X 2 + Y 2 2 + 2E U 2 + V 2 . X 2 + Y 2 + 4E (U X + V Y ) 2 . (A.8)
Based on the definition of the QPSK signal, (U 2 + V 2 ) is a random discrete variable taking on two equiprobable values, 1/2 • (1 + (α + β) 2 ) and 1/2 • (1 + (αβ) 2 ). The first term of (A.8) can then be derived as

E U 2 + V 2 2 = 1 4 1 + α 2 + β 2 2 + 4α 2 β 2 . (A.9)
The second term of (A.8) is calculated using the fact that all the moments of a zero mean Gaussian variable can be derived from its variance [START_REF] Magnus | Formulas and Theorems for the Special Functions of Mathematical Physics[END_REF]. More specifically, the fourth order moments of (X, Y ) can be written as (U 2 + V 2 ) and (Y 2 + X 2 ) are independent random variables. The third term of (A.8) is then obtained

     E X 4 = 3σ 4 E Y 4 = 3 α 2 + β 2 2 σ 4 E X 2 Y 2 = E
E U 2 + V 2 . X 2 + Y 2 = E U 2 + V 2 • E X 2 + Y 2 = σ 2 2 1 + α 2 + β 2 2
. (A.12)

Due to the independence property between the variables (U, V ) and (X, Y ), the last term of (A.8) can be reformulated as

E (U X + V Y ) 2 = E U 2 E X 2 + 2E [U V ] • E [XY ] + E V 2 • E Y 2 . (A.13)
The last term is expressed as

E (U X + V Y ) 2 = α 2 + 1 + α 2 + β 2 2 2 • σ 2 . (A.14)
From (A.9), (A.11), (A.12) and (A.14), the variance of |r| 2 is finally derived (as in Eq. ( 8))

Var |r| 2 = α 2 β 2 + 2 1 + 2α

Appendix B

If φ 0 = arg max φvar D (wo) (φ var ) or tan φ 0 = -sin φ mis , we evaluate the first derivative of D(φ) at φ = φ 0 . To do that, the expression of D in Eq. ( 9) is represented as follows By inspecting Eq. (B.2), it can be observed that the extremum of D corresponds to the extremum of F S (φ). Instead of evaluating the first derivative of D at φ = φ 0 , we can equivalently investigate the first derivative of F S (φ) at φ = φ 0 . In order to simplify the expression, the following variables and its corresponding derivative at φ = φ 0 are established

D = Z 1 + α 2 + β 2 2
       u = α 2 , v = β 2 u 0 = α 2 φ=φ 0 = 0, v 0 = β 2
dφ φ=φ 0 = 2S • K (1 + v 0 ) 2 , (B.6)
in which K is calculated by

K = 2v 0 1 + v 0 -v 0 + v 0 . (B.7)
Due to the fact that tan φ 0 = -sin φ mis and from the definition of β in Eq. ( 6), the values of v 0 and v 0 can be deduced as follows v 0 = cos 2 φ mis 1+sin 2 φ mis v 0 = 2cos 2 φ mis sin φ mis 1+sin 2 φ mis .

(B.8)

Figure 1 :

 1 Figure 1: Structure of the coherent receiver, including a 90 • hybrid, in the presence of the phase imbalance (or mismatch) φ mis . S: signal input, LO: local oscillator input.

Figure 2 :

 2 Figure 2: Analytical prediction of the estimated phase imbalance values, φmis for different predefined phase imbalance values, φ mis using the proposed MSEM method and their corresponding estimated phase errors, φmis -φ mis . φ var is the IQ compensation phase value in the range [-30 • , 30 • ].

Figure 3 :

 3 Figure 3: Calculation of the proposed metric, D, as a function of the variable IQ compensation phase φ var for a pre-set phase imbalance of -15 • and different received samples SNR values.

Figure 4 :

 4 Figure 4: Estimated phase imbalance under the impact of AWGN. Dashed lines: pre-defined phase imbalances of 10 • , 20 • and 30 • . Solid lines: estimated phase imbalance values.

Figure 5 :

 5 Figure 5: (a) Experimental setup for characterizing the quadrature imbalance compensation in a QPSK coherent system. (b) Dual drive IQ modulator structure. CW: continuous wave laser. AWG: arbitrary waveform generator. VOA: variable optical attenuator. EDFA: erbium-doped fiber amplifier. S: signal. LO: local oscillator. MZM: Mach-Zehnder modulator.

Figure 6 :

 6 Figure 6: Experimental QPSK constellations at 12 dB OSNR: (a) without IQ imbalance, (b) with IQ imbalance of 15 • , (c) recovered using the GSOP method, (d) recovered using the proposed MSEM method.

Figure 7 :

 7 Figure7: BER measurement as a function of OSNR (in 0.1 nm) for phase misalignments of 0 • and 17 • without any correction and with compensation using the GSOP and MSEM methods.

Figure 8 :

 8 Figure 8: EVM calculation under the impact of IQ imbalance without (w/o) and with compensation by the GSOP and MSEM methods.

n 2 C 4 . (A. 10 )

 2410 (αn C + βn S ) 2 = 3α 2 + β 2 σTherefore, the second term of (A.8) is given byE X 2 + Y 2 2 = 3 + 2 3α 2 + β 2 + 3 α 2 + β 2 2 σ 4 . (A.11)

α 2 β 2 -(1+α 2 +β 2 ) 2 .

 22 2Sβ 2 + S(1 + α 2 + β 2 ) 2 = Z S + β 2 (α 2 -2S) (B.1) where Z = (1/2 + σ 2 ) 2 and S = 2σ 2 (1 + σ 2 ). If we call F S (φ) = β 2 (α 2 -2S) (1+α 2 +β 2 ) 2 ,Eq. (B.1) can be re-written by D = Z S + F S (φ) . (B.2)

3 )

 3 The expression of F S (φ) is now given byF S (φ) = v (u -2S) (1 + u + v) 2 (B.4)and its first derivative at φ = φ 0 can be establishedd (F S (φ)) dφ φ=φ 0 = v 0 (u 0 -2S) + v 0 (u 0 -2S) (1 + u 0 + v 0 ) 2 -2v 0 (u 0 -2S) (1 + u 0 + v 0 ) 3 (B.5)or equivalently, d (F S (φ))

Table 1 :

 1 Comparison of hardware complexity Methods Real-addersReal-multipliers Square-root operators

  2 + α 2 + β 2 2 • 1 + α 2 • σ 2 . (A.15)

Acknowledgements

This work was supported by the French National Research Agency (ANR) in the frame of the OCELOT project (ref. ANR-10-VERS-0015), the Contrat de plan Etat-Région Ponant and the French Ministry of Research.

Appendix A

Based on the definition of the new metric in (4), the squared modulus of r can be written by

in which the variables U , X, V and Y are defined by

We first calculate the expected value of |r| 2 . From (A.1), the expectation of r 2 is calculated as follows

It must be noted that the expected value of U , V , X and Y is zero. On the other hand, U and X, V and Y are uncorrelated so that the last two terms of (A.3) are equal to 0. Moreover, φ S is uniformly distributed over the four values (2k + 1) • π/4, k = 0, 1, 2, 3. Using the quantities defined in (A.2), the variances of U , V and their co-variance can be retrieved as

As mentioned, n C and n S are random Gaussian variables with the same variance σ 2 . The variances of X, Y and their covariance are

(A.5) Substituting (B.8) into (B.7), K can be obtained as K = cos 2 φ mis sin φ mis (sin φ mis -2) 1 + sin 2 φ mis . (B.9)

From (B.6) and (B.9), it can be concluded that K is different from 0 and hence the first derivative of F S (φ) at φ = φ 0 is also not equal to 0 whenever S = 0. As a consequence, the first derivative of D(φ) at φ = φ 0 differs from 0, so the dependence of the maximum value of D on the SNR of the received samples is confirmed.