Rémi Cuingnet
email: remi.cuingnet@imed.jussieu.fr

Marie Chupin
email: marie.chupin@upmc.fr

Habib Benali
email: habib.benali@imed.jussieu.fr

Olivier Colliot
email: olivier.colliot@upmc.fr

Spatial and anatomical regularization of SVM for brain image analysis

Support vector machines (SVM) are increasingly used in brain image analyses since they allow capturing complex multivariate relationships in the data. Moreover, when the kernel is linear, SVMs can be used to localize spatial patterns of discrimination between two groups of subjects. However, the features' spatial distribution is not taken into account. As a consequence, the optimal margin hyperplane is often scattered and lacks spatial coherence, making its anatomical interpretation difficult. This paper introduces a framework to spatially regularize SVM for brain image analysis. We show that Laplacian regularization provides a flexible framework to integrate various types of constraints and can be applied to both cortical surfaces and 3D brain images. The proposed framework is applied to the classification of MR images based on gray matter concentration maps and cortical thickness measures from 30 patients with Alzheimer's disease and 30 elderly controls. The results demonstrate that the proposed method enables natural spatial and anatomical regularization of the classifier.

Introduction

Brain image analyses have widely relied on univariate voxel-wise analyses, such as voxel-based morphometry (VBM) for structural MRI [START_REF] Ashburner | Voxel-based morphometry-the methods[END_REF]. In such analyses, brain images are first spatially registered to a common stereotaxic space, and then mass univariate statistical tests are performed in each voxel to detect significant group differences. However, the sensitivity of theses approaches is limited when the differences are spatially complex and involve a combination of different voxels or brain structures [START_REF] Davatzikos | Why voxel-based morphometric analysis should be used with great caution when characterizing group differences[END_REF]. Recently, there has been a growing interest in support vector machines (SVM) methods [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF][START_REF] Schölkopf | Learning with Kernels[END_REF] to overcome the limits of these univariate analyses. Theses approaches allow capturing complex multivariate relationships in the data and have been successfully applied to the individual classification of a variety of neurological conditions [START_REF] Lao | Morphological classification of brains via high-dimensional shape transformations and machine learning methods[END_REF][START_REF] Fan | COMPARE: classification of morphological patterns using adaptive regional elements[END_REF][START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF][START_REF] Vemuri | Alzheimer's disease diagnosis in individual subjects using structural MR images: validation studies[END_REF]. Moreover, the output of the SVM can also be analyzed to localize spatial patterns of discrimination, for example by drawing the coefficients of the optimal margin hyperplane (OMH) -which, in the case of a linear SVM, live in the same space as the MRI data [START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF][START_REF] Vemuri | Alzheimer's disease diagnosis in individual subjects using structural MR images: validation studies[END_REF]. However, one of the problems with analyzing directly the OMH coefficients is that the corresponding maps are scattered and lack spatial coherence. This makes it difficult to give a meaningful interpretation of the maps, for example to localize the brain regions altered by a given pathology.

In this paper, we address this issue by proposing a framework to introduce spatial consistency into SVMs by using regularization operators. Section 2 provides some background information on SVMs and regularization operators. We then show that the regularization operator framework provides a flexible approach to model different types of proximity (section 3). Section 4 presents the first type of regularization, which models spatial proximity, i.e. two features are close if they are spatially close. We then present in section 5 a more complex type of constraint, called anatomical proximity. In the latter case, two features are considered close if they belong to the same brain network; for instance two voxels are close if they belong to the same anatomical or functional region or if they are anatomically or functionally connected (based on fMRI networks or white matter tracts). Finally, in section 6, the proposed framework is illustrated on the analysis of MR images using gray matter concentration maps and cortical thickness measures from 30 patients with AD and 30 elderly controls from the ADNI database (www.adni-info.org).

Priors in SVM

In this section, we first describe the neuroimaging data that we consider in this paper. Then, after some background on SVMs and on how to add prior knowledge in SVMs, we describe the framework of regularization operators.

Brain imaging data

In this contribution, we consider any feature computed either at each voxel of a 3D brain image or at any vertex of the cortical surface. Typically, for anatomical studies, the features could be tissue concentration maps such as gray matter (GM) or white matter (WM) for the 3D case or cortical thickness maps for the surface case. The proposed methods are also applicable to functional or diffusion weighted MRI. We further assume that 3D images or cortical surfaces were spatially normalized to a common stereotaxic space (e.g. [START_REF] Ashburner | A fast diffeomorphic image registration algorithm[END_REF]) as in many group studies or classification methods [START_REF] Lao | Morphological classification of brains via high-dimensional shape transformations and machine learning methods[END_REF][START_REF] Fan | COMPARE: classification of morphological patterns using adaptive regional elements[END_REF][START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF][START_REF] Vemuri | Alzheimer's disease diagnosis in individual subjects using structural MR images: validation studies[END_REF][START_REF] Querbes | Early diagnosis of Alzheimer's disease using cortical thickness: impact of cognitive reserve[END_REF].

Let V be the domain of the 3D images or surfaces. v will denote an element of V (i.e. a voxel or a vertex). Thus, X = R V , together with the canonical dot product will be the input space.

Let x s ∈ X be the data of a given subject s. In the case of 3D images, x s can be considered in two different ways: (i) as an element of R d where d denotes the number of voxels, (ii) as a real-valued function defined on a compact subset of R 3 . Both finite and continuous viewpoints will be studied in this paper because they allow different types of regularization. Similarly, in the surface case, x s can be viewed either as an element of R d where d denotes the number of vertices or as a real-valued function on a 2-dimensional compact Riemannian manifold.

We consider a group of N subjects with their corresponding data (x s ) s∈[1,N ] ∈ X N . Each subject is associated with a group (y s ) s∈[1,N ] ∈ {-1, 1} N (typically his diagnosis, i.e. diseased or healthy).

Linear SVM

The linear SVM solves the following optimization problem [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF][START_REF] Schölkopf | Learning with Kernels[END_REF][START_REF] Shawe-Taylor | Kernel methods for pattern analysis[END_REF]:

w opt , b opt = arg min w∈X ,b∈R N s=1 l hinge (y s [ w, x s + b]) + λ w 2 (1)
where λ ∈ R + is the regularization parameter and l hinge the hinge loss function defined as:

l hinge : u ∈ R → max(0, 1 -u).
With a linear SVM, the feature space is the same as the input space. Thus, when the input features are the voxels of a 3D image, each element of w opt = (w opt v ) v∈V also corresponds to a voxel. Similarly, for the surface-based methods, the elements of w opt can be represented on the vertices of the cortical surface. To be anatomically consistent, if v (1) ∈ V and v (2) ∈ V are close according to the topology of V, their weights in the SVM classifier, w opt v (1) and w opt v (2) respectively, should be similar. In other words, if v (1) and v (2) correpond to two neighboring regions, they should have a similar role in the classifier function. However, this is not guaranteed with the standard linear SVM (as for example in [START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF]) because the regularization term is not a spatial regularization. The aim of the present paper is to propose methods to ensure that w opt is spatially regularized.

How to include priors in SVM

To spatially regularize the SVM, one has to include some prior knowledge on the proximity of features. In the literature, three main ways have been considered in order to include priors in SVMs.

In an SVM, all the information used for classification is encoded in the kernel. Hence, the first way to include prior is to directly design the kernel function [START_REF] Schölkopf | Learning with Kernels[END_REF]. But this implies knowing a metric on the input space X consistent with the prior knowledge.

Another way is to force the classifier function to be locally invariant to some transformations. This can be done: (i) by directly engineering a kernel which leads to locally invariant SVM, (ii) by generating artificially transformed examples from the training set to create virtual support vectors (virtual SV), (iii) by using a combination of both these approaches called kernel jittering [START_REF] Decoste | Training invariant support vector machines[END_REF][START_REF] Schölkopf | Incorporating invariances in support vector learning machines[END_REF][START_REF] Schölkopf | Prior knowledge in support vector kernels[END_REF]. But the main difficulty here is how to define the transformations to which we would like the kernel to be invariant.

The last way is to consider SVM from the regularization viewpoint [START_REF] Smola | On a kernel-based method for pattern recognition, regression, approximation, and operator inversion[END_REF][START_REF] Schölkopf | Learning with Kernels[END_REF]. The idea is to force the classifier function to be smooth with respect to some criteria. This is the viewpoint which is adopted in this paper.

Regularization operators

Our aim is to introduce a spatial regularization on the classifier function of the SVM which can be written as sgn (f (x s ) + b) where f ∈ R X . This is done through the definition of a regularization operator P on f . Following [START_REF] Smola | On a kernel-based method for pattern recognition, regression, approximation, and operator inversion[END_REF][START_REF] Schölkopf | Learning with Kernels[END_REF], P is defined as a linear map from a space

F ⊂ R X into a dot product space (D, •, • D ). G : X × X → R is a Green's function of a regularization operator P iff: ∀f ∈ F, ∀x ∈ X , f (x) = P (G(x, •), P (f ) D (2)
If P admits at least a Green's function called G, then G is a positive semi-definite kernel and the minimization problem:

f opt , b opt = arg min f ∈F ,b∈R N s=1 l hinge (y s [f (x s ) + b]) + λ P (f ) 2 D (3)
is equivalent to the SVM minimization problem with kernel G.

Since in linear SVM, the feature space is the input space, f lies in the input space. Therefore, the optimisation problem (3) is very convenient to include spatial regularization on f via the definition of P . Note that, usually, F is a Reproducing Kernel Hilbert Space (RKHS) with kernel K and D = F. Hence, if P is bounded, injective and compact, P admits a Green's function G = (P † P ) -1 K where P † denotes the adjoint of P .

One has to define the regularization operator P so as to obtain the suitable regularization for the problem.

Laplacian regularization

Spatial regularization requires the notion of proximity between elements of V. This can be done through the definition of a graph in the discrete case or a metric in the continuous case. In this section, we propose spatial regularizations based on the Laplacian for both of these proximity models. This penalizes the high-frequency components with respect to the topology of V.

Graphs

When V is finite, weighted graphs are a natural framework to take spatial information into consideration. Voxels of a brain image can be considered as nodes of a graph which models the voxels' proximity. This graph can be the voxel connectivity (6, 18 or 26) or a more sophisticated graph.

We chose the following regularization operator:

P : w * ∈ F = L(R V , R) → e 1 2 βL w * ∈ F (4)
where L denotes the graph Laplacian [START_REF] Chung | Spectral Graph Theory[END_REF] and w * the dual vector of w. β controls the size of the regularization. The optimization problem then becomes:

(w opt , b opt ) = arg min w∈X ,b∈R N s=1 l hinge (y s [ w, x s + b]) + λ e 1 2 βL w 2 (5)
Such a regularization exponentially penalizes the high-frequency components and thus forces the classifier to consider as similar voxels highly connected according to the graph adjacency matrix. According to the previous section, this new minimization problem ( 5) is equivalent to an SVM optimization problem. The new kernel K β is given by:

K β (x 1 , x 2 ) = x T 1 e -βL x 2 (6
) This is a heat or diffusion kernel on a graph. Our approach differs from the diffusion kernels introduced by Kondor et al. [START_REF] Kondor | Diffusion kernels on graphs and other discrete input spaces[END_REF] because the nodes of the graph are the features, here the voxels, whereas in [START_REF] Kondor | Diffusion kernels on graphs and other discrete input spaces[END_REF], the nodes were the objects to classify. Laplacian regularization was also used in satellite imaging [START_REF] Gómez-Chova | Semi-supervised image classification with Laplacian support vector machines[END_REF] but, again, the nodes were the objects to classify. Our approach can also be considered as a spectral regularization on the graph [START_REF] Smola | Kernels and regularization on graphs[END_REF]. To our knowledge, such spectral regularization has not been applied to brain images but only to the classification of microarray data [START_REF] Rapaport | Classification of microarray data using gene networks[END_REF].

Compact Riemannian manifolds

In this paper, when V is continuous, it can be considered as a 2-dimensional (e.g. surfaces) or a 3-dimensional (e.g. 3D Euclidean or more complex) compact Riemannian manifold. The metric then models the notion of proximity. On such spaces, the heat kernel exists [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF][START_REF] Lafferty | Diffusion kernels on statistical manifolds[END_REF]. Therefore, the Laplacian regularization presented in the previous paragraph can be extended to compact Riemannian manifolds [START_REF] Lafferty | Diffusion kernels on statistical manifolds[END_REF]. Similarly to the graphs, we chose the following regularization operator:

P : w * ∈ F = L(R V , R) → e 1 2 β∆ w * ∈ F (7)
where ∆ denotes the Laplace-Beltramin operator. The optimization problem is also equivalent to an SVM optimization problem with kernel K β (x 1 , x 2 ) = x T 1 e -β∆ x 2 . Note the difference between our approach and that of Laferty and Lebanon [START_REF] Lafferty | Diffusion kernels on statistical manifolds[END_REF]. In our case, the points of the manifolds are the features, whereas in [START_REF] Lafferty | Diffusion kernels on statistical manifolds[END_REF], they were the objects to classify.

In sections 4 and 5, we present different types of proximity models which correspond to different types of graphs or distances.

Spatial proximity

In this section, we consider the case of regularization based on spatial proximity, i.e. two voxels (or vertices) are close if they are spatially close.

The 3D case

When V are the image voxels (discrete case), the simplest option to encode the spatial proximity is to use the image connectivity (e.g. 6-connectivity) as a regularization graph. Similarly, when V is a compact subset of R 3 (continuous case), the proximity is encoded by a Euclidean distance. In both cases, this is equivalent to pre-process the data with a Gaussian smoothing kernel with standard deviation σ = √ β [START_REF] Kondor | Diffusion kernels on graphs and other discrete input spaces[END_REF].

However, smoothing the data with a Gaussian kernel would mix gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF). Instead, we propose a graph which takes into consideration both the spatial localization and the tissue types. Based on tissue probability maps, in each voxel v, we have the set of probabilities p v that this voxel belongs to GM, WM or CSF. We considered the following graph. Two voxels are connected if and only if they are neighbors in the image (6-connectivity). The weight a u,v of the edge between two connected voxels u and v is a u,v = e -d χ 2 (pu,pv) 2 /(2σ 2 ) , where d χ 2 is the χ 2 -distance between two distributions. We chose beforehand σ equal to the standard deviation of d χ 2 (p u , p v ).

To compute the kernel, we computed e -βL x s for each subject s in the training set by scaling the Laplacian and using the Taylor series expansion.

The surface case

The connectivity graph is not directly applicable to surfaces. Indeed, the regularization would then strongly depend on the mesh used to discretize the surface. This shortcoming can be overcome by reweighing the graph with conformal weights. In this paper, we chose a different approach by adopting the continuous viewpoint: we consider the cortical surface as a 2-dimensional Riemannian manifold and use the regularization operator defined by equation [START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF]. Indeed, the Laplacian is an intrinsic operator and does not depend on the chosen surface parameterization. The heat kernel has already been used for cortical smoothing for example in [START_REF] Andrade | Detection of fMRI activation using cortical surface mapping[END_REF][START_REF] Cachia | A primal sketch of the cortex mean curvature: a morphogenesis based approach to study the variability of the folding patterns[END_REF][START_REF] Chung | Heat kernel smoothing and its application to cortical manifolds[END_REF][START_REF] Chung | Cortical thickness analysis in autism with heat kernel smoothing[END_REF]. We will therefore no detail this part. We used the implementation described in [START_REF] Chung | Cortical thickness analysis in autism with heat kernel smoothing[END_REF].

Anatomical proximity

In this section, we consider a different type of proximity, which we call anatomical proximity. Two voxels are considered close if they belong to the same brain network. For example, two voxels can be close if they belong to the same anatomical or functional region (defined for example by a probabilistic atlas). This can be seen as a "short-range" connectivity. Another example is that of "long-range" proximity which models the fact that distant voxels can be anatomically (through white matter tracts) or functionally connected (based on fMRI networks).

We first focus on the discrete case. The presented framework can be used either for 3D images or surfaces and computed very efficiently. However, such an efficient implementation was obtained at the cost of the spatial proximity. Therefore, we then show a continuous formulation which enables to consider both spatial and anatomical proximity.

On graphs: atlas and connectivity

Let (A 1 , • • • , A R )
be the R regions of interest (ROI) of an atlas and p(v ∈ A r ) the probability that the voxel v belongs to region A r . Then the probability that two voxels v (i) and v (j) belong to the same region is: R r=1 p v (i) , v (j) ∈ A 2 r . We assume that if v (i) = v (j) then: p v (i) , v (j) ∈ A 2 r = p v (i) ∈ A r p v (j) ∈ A r . Let E ∈ R d×R be the right stochastic matrix defined by: E i,r = p v (i) ∈ A r . Then, for i = j, the (i, j)-th entry of the adjacency matrix EE t is the probability that the voxels v (i) and v (j) belong to the same regions.

For "long-range" connections (structural or functional), one can consider an R-by-R matrix C with the (r 1 , r 2 )-th entry being the probability that A r1 and A r2 are connected. Then the adjacency matrix becomes: ECE t . We considered the normalized Laplacian L [START_REF] Chung | Spectral Graph Theory[END_REF], to be sure that the two terms commute:

L = I d -D -1 2 ECE t D -1 2 (8)
where D is a diagonal matrix. Hence, if CE t D -1 E is not singular, we have:

e -β L = e -β I d + D -1 2 E(e βCE t D -1 E -I R )(CE t D -1 E) -1 CE t D -1 2 (9)
The computation requires only the computation of D -1 2 , which is done efficiently since D is a diagonal matrix, and the computation of inverse and the matrix exponential of an R-by-R matrix, which is also efficient since R ∼ 10 2 . This method can be directly applied to both 3D images and cortical surfaces. Unfortunately, the efficient implementation was obtained at the cost of the spatial proximity. The next section presents a combination of anatomical and spatial proximity using the continuous viewpoint.

On statistical manifolds

In this section, the goal is to take into account various prior informations such as tissue information, atlas information and spatial proximity. We first show that this can be done by considering the images or surfaces as statistical manifolds together with the Fisher metric. We then give some details about the computation of the kernel.

Fisher metric We assume that we are given an anatomical or a functional atlas A composed of R regions: {A r } r=1•••R . Similarly, T = {T GM , T WM , T CSF } denotes the set of brain tissues. In each point v ∈ V, we have a probability distribution p atlas (•|v) ∈ R T ×A which informs about the tissue type and the atlas region in v. Without any loss of generality, one can assume that the tissue information is encoded in the atlas. Therefore, we consider the probability p atlas (•|v) ∈ R A . We also consider a probability distribution p loc (•|v) ∈ R V which encodes the spatial proximity. A simple example is p loc (•|v) ∼ N (v, σ 2 loc ). Therefore, we consider the probability family:

M = p(•|v) ∈ R A×V
v∈V where p(•|v) = p atlas (•|v)p loc (•|v). A natural way to encode proximity on M is to use the Fisher metric as in [START_REF] Lafferty | Diffusion kernels on statistical manifolds[END_REF]. With some smoothness assumption about p, M together with this metric is a compact Riemannian manifold [START_REF] Amari | Differential Geometry in Statistical Inference[END_REF]. For clarity, we present this framework only for 3D images but it could be applied to cortical surfaces with minor changes. The metric tensor g is then given for all v ∈ V by:

g ij (v) = E v ∂ log p(•|v) ∂v i ∂ log p(•|v) ∂v j , 1 ≤ i, j ≤ 3 ( 10
)
If we further assume that p loc (•|v) is isotropic we have:

g ij (v) ≈ g atlas ij (v) + δ ij u∈V p loc (u|v) ∂ log p loc (u|v) ∂v i 2 du ( 11
)
where δ ij is the Kronecker delta and g atlas is the metric tensor when p(•|v) = p atlas (•|v). When

p loc (•|v) ∼ N (v, σ 2 loc I 3 ), we have: g ij (v) ≈ g atlas ij (v) + δ ij σ 2 loc .
Computing the kernel Once the notion of proximity is defined, one has to compute the kernel matrix. The computation of the kernel matrix requires the computation of e -β∆ x s for all the subjects of the training set. The eigendecomposition of the Laplace-Beltrami operator is intractable since the number of voxels in a brain images is about 10 6 . Hence e -β∆ x s is considered as the solution at t = β of the heat equation with the Dirichlet homogeneous boundary conditions:

∂u ∂t -∆u = 0 u(t = 0) = x s (12)
The Laplace-Beltrami operator is given by [START_REF] Jost | Riemannian geometry and geometric analysis[END_REF]:

∆u = 1 √ det g 3 j=1 ∂ ∂v j 3 i=1 h ij det g ∂u ∂v i
where h is the inverse tensor of g.

To solve equation [START_REF] Decoste | Training invariant support vector machines[END_REF], one can use a variational approach [START_REF] Druet | Blow-up theory for elliptic PDEs in Riemannian geometry[END_REF]. We used the rectangular finite elements in space and the explicit finite difference scheme for the time discretization. ∆ x and ∆ t denote the space step and the time step respectively. ∆ x is fixed by the MRI spatial resolution. ∆ t is then chosen so as to respect the Courant-Friedrichs-Lewy (CFL) condition, which can be written in this case as:

∆ t ≤ 2(max λ i ) -1
, where λ i are the eigenvalues of the general eigenproblem: KU = λMU with K the stiffness matrix and M the mass matrix. To compute the optimal time step ∆ t , we estimated the largest eigenvalue with the power iteration method.

6 Experiments and results

Material

Subjects and MRI acquisition Data were obtained from the Alzheimer's Disease Neuroimaging Initiative (ADNI) database 1 . The Principal Investigator of this initiative is Michael W. Weiner, M.D., VA Medical Center and University of California -San Francisco.For up-to-date information see www.adni-info.org. We studied 30 patients with probable AD (age± standard-deviation (SD) = 74±4, range = 60-80 years, mini-mental score (MMS) = 23±2) and 30 elderly controls (age± SD = 73±4, range = 60-80, MMS = 29±1) which were selected from the ADNI database according to the following criteria. Subjects were excluded if their scan revealed major artifacts or gross structural abnormalities of the white matter, for it makes the tissue segmentation step fail. 80-year-old subjects or older were also excluded. The MR scans are T1-weighted MR images. MRI acquisition was done according to the ADNI acquisition protocol in [START_REF] Jack | The Alzheimer's Disease Neuroimaging Initiative (ADNI): MRI methods[END_REF].

Features extraction For the 3D image analyses, all T1-weighted MR images were segmented into gray matter (GM), white matter (WM) and cerebrospinal fluid (CSF) using the SPM5 (Statistical Parametric Mapping, London, UK) unified segmentation routine [START_REF] Ashburner | Unified segmentation[END_REF] and spatially normalized with DARTEL [START_REF] Ashburner | A fast diffeomorphic image registration algorithm[END_REF]. The features are the GM probability maps in the MNI space. For the surface-based analyses, the features are the cortical thickness values at each vertex of the cortical surface. Cortical thickness measures were performed with Freesurfer (Massachusetts General Hospital, Boston, MA).

Proposed experiments

As an illustration of the method, we present the results of the AD versus controls analysis. We present the maps associated to the optimal margin hyperplane (OMH). The classification function obtained with a linear SVM is the sign of the inner product of the features with w opt , a vector orthogonal to the OMH [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF][START_REF] Schölkopf | Learning with Kernels[END_REF]. Therefore, if the absolute value of the i th component of w opt , |w opt i |, is small compared to the other components (|w opt j |) j =i , the i th feature will have a small influence on the classification. Conversely, if |w opt i | is relatively large, the i th feature will play an important role in the classifier. Thus the optimal weights w opt allow us to evaluate the anatomical consistency of the classifier. In all experiments, the C parameter of the SVM was fixed to one (λ = 1 2N C [START_REF] Schölkopf | Learning with Kernels[END_REF]).

Results: spatial proximity

In this section, we present the results for the spatial proximity in the 3D case (method presented in section 4.1). Due to space limitations, the surface case is not presented. Fig. 1(a) presents the OMH when no spatial regularization is performed. Fig. 1(b) shows the results with spatial proximity but without tissue probability maps. w becomes smoother and spatially consistent. However it mixes tissues and does not respect the topology of the cortex. For instance, it mixes tissues of the temporal lobe with tissues of the frontal and parietal lobes. The results with both spatial proximity and tissue maps are shown on Fig. 1(c). The OMH is much more consistent with the brain anatomy. β controls the size of the spatial regularization and was chosen to be equivalent to a 4mm-FWHM of the Gaussian smoothing. The classification accuracy was estimated by a leave-one-out cross validation.

The classifiers were able to distinguish AD from CN with similar accuracies (83% with no spatial priors and 85% with spatial priors).

Results: anatomical proximity

In this section, we present the results for the anatomical proximity. We first present the discrete surface case. The discrete 3D case leads to comparable results but is omitted here due to space limitations. We then present the continuous 3D case. Extension to surfaces is left for future work.

Discrete case For the discrete case, we used "short-range" proximity, defined by the cortical atlas of Desikan et al. [START_REF] Desikan | An automated labeling system for subdividing the human cerebral cortex on MRI scans into gyral based regions of interest[END_REF] with binary probabilities. We tested different values for β = 0, 1, • • • , 5.

The accuracies ranged between 80% and 85%. The highest accuracy was reached for β = 3. The optimal SVM weights w are shown on Fig. 2. When no regularization has been carried out, they are noisy and scattered (Fig. 2 (a)). When the amount of regularization is increased, voxels of a same region tend to be considered as similar by the classifier (Fig. 2(b-d)). Note how the anatomical coherence of the OMH varies with β.

Continuous case

We then present the results of the 3D continuous case (section 5.2). The atlas information used was only the tissue types. We chose σ loc = 10mm for the spatial confidency. β was chosen to be equivalent to a 4mm-FWHM of the Gaussian smoothing. The classifier reached 87% accuracy. The optimal SVM weights w are shown on Fig. 1(d). The tissue knowledge enables the classifier to be more consistent with the anatomy. For instance, note the difference with the Gaussian smoothing (Fig. 1(b)) and how the proposed method avoids mixing the temporal lobe with the parietal and frontal lobes.

Discussion

In this contribution, we proposed to use regularization operators to add spatial consistency to SVMs for brain image analysis. We show that this provides a flexible approach to model different types of proximity between the features. We proposed derivations for both 3D image features, such as tissue maps, or surface characteristics, such as cortical thickness. We considered two different types of formulations: a discrete viewpoint in which the proximity is encoded via a graph, and a continuous viewpoint in which the data lies on a Riemannian manifold. In particular, the latter viewpoint is useful for surface cases because it overcomes problems due to surface parameterization. This paper introduced two different types of proximity. We first considered the case of regularization based on spatial proximity, which results in spatially consistent OMH making their anatomical interpretation more meaningful. We then considered a different type of proximity which allows modeling higherlevel knowledge, which we call anatomical proximity. In this model, two voxels are considered close if they belong to the same brain network. For example, two voxels can be close if they belong to the same anatomical region. This can be seen as a "short-range" connectivity. Another example is that of "long-range" proximity which models the fact that distant voxels can be anatomically connected, through white matter tracts, or functionally connected, based on fMRI networks.

Preliminary evaluation was performed on 30 patients with AD and 30 age-matched controls. The results demonstrate that the proposed approaches allow obtaining spatially and anatomically coherent discrimination patterns. In particular, the obtained hyperplanes are largely consistent with the neuropathology of AD, with highly discriminant features in the medial temporal lobe, as well as lateral temporal, parietal associative and frontal areas. As for the classification results, they were comparable to those reported in the literature for AD classification (e.g. [START_REF] Lao | Morphological classification of brains via high-dimensional shape transformations and machine learning methods[END_REF][START_REF] Vemuri | Alzheimer's disease diagnosis in individual subjects using structural MR images: validation studies[END_REF][START_REF] Klöppel | Automatic classification of MR scans in Alzheimer's disease[END_REF]). The use of regularization did not substantially improve the accuracy. However, the most important point is that the proposed approach makes the results more consistent with the anatomy, making their interpretation more meaningful.

Finally, it should be noted that the proposed approach is not specific to structural MRI, and can be applied to other pathologies and other types of data (e.g. functional or diffusion-weighted MRI).

Figure 1 :Figure 2 :

 12 Figure 1: Normalized w coefficients: (a) no spatial prior, (b) spatial proximity: FWHM=4mm, (c) spatial proximity and tissues: FWHM∼4mm, (d) Fisher metric using tissue maps. -0.5 0 0 +0.5

www.loni.ucla.edu/ADNI

Acknowledgments

This work was supported by ANR (project HM-TC, number ANR-09-EMER-006). Data collection and sharing for this project was funded by the Alzheimer's Disease Neuroimaging Initiative (ADNI; Principal Investigator: Michael Weiner; NIH grant U01 AG024904). ADNI data are disseminated by the Laboratory of Neuro Imaging at the University of California, Los Angeles.


Keywords: Antenna array, dual circular polarization, dual frequency, annular slot, sequential rotation technique

A novel Ka-band dual-band dual circularly polarized antenna array is presented in this letter. A dual-band antenna with left-hand circular polarization for the Ka-band downlink frequencies and right-hand circular polarization for the Ka-band uplink frequencies is realized with compact annular ring slots. By applying the sequential rotation technique, a 2×2 subarray with good performance is obtained. This paper describes the design process and presents simulation and measurement results.

I. INTRODUCTION

ecause satellite communication (SatCom) systems having large capacities are in demand, new SatCom antennas using the Ka-band are being investigated. This higher operating frequency provides at the same time high data rate links and miniaturized antenna systems [START_REF] Marcellini | Future developments trend for Ku and Ka antenna for satcom on the move[END_REF]. In terms of choice of polarization, circularly polarized (CP) microstrip antennas [START_REF] Uhl | Quadrature-modulated circular microstrip patch antenna for phased arrays[END_REF][START_REF] Gao | Low-cost broadband circularly polarized printed antennas and array[END_REF][START_REF] Chung | HTS microstrip antenna array for circular polarization with cryostat[END_REF][START_REF] Nasimuddin | Wideband circularly polarized microstrip antenna array using a new single feed network[END_REF] are most attractive because they can facilitate orientation between transmitter and receiver and are highly suitable for mobility, weather penetration, and reduction of multipath reflections [START_REF] Garg | Microstrip Antenna Design Handbook[END_REF]. The proposed antenna system therefore exhibits dual band functionality with dual-circularly polarization operation to provide an independent single beam with a large steering range in each band: uplink from 30 GHz to 31 GHz with Right-Hand CP and downlink from 20 GHz to 21 GHz with Left-Hand CP. Recent developments mainly use parabolic antennas with mechanical beam coverage [START_REF] Khalil | Satcom-On-The-Move Presentation[END_REF][START_REF] Kot | Antenna concepts for Kaband SATCOM on the move[END_REF][START_REF] Kot | Compact Shaped Dual-Reflector System For Military Ka-band SATCOM on the move[END_REF][START_REF] Bayer | Land-Mobile Ka-Band Satcom Tracking Antenna Employing a Printed Circuit Board Based Multimode Monopulse Feed[END_REF] or multilayer configurations [START_REF] Meniconi | 3D integration technologies for a planar dual band active array in Ka-band[END_REF][START_REF] Greco | A Dual-Band, Dual-Polarized Array Element for Ka Band Satcom on the Move Terminals[END_REF] with a significant cost. Microstrip antennas are widely preferred especially when profile, size, weight, compatibly with other integrated circuitry and costs are of main concern. In recent years, only a few investigations of Ka-band SatCom antennas operating under circular-polarization have been carried out. In [START_REF] Chen | Development of a Ka-band wideband circularly polarized 64-element microstrip antenna array with double application of the sequential rotation feeding technique[END_REF], a multiple series-parallel sequential rotation feeding network for 64 elements is presented but only for a single band. Dual-CP operation has been demonstrated using a mechanical polarization screen [START_REF] Lenormand | Antenne mobile large bande `a commutation de polarisation[END_REF] or by using the stacked patch technique [START_REF] Rohrdantz | A dualfrequency and dual-polarized patch antenna at Ka-band[END_REF].

In comparison to this design, as introduced in [START_REF] Amendola | Annular ring slot radiating element for integrated millimeter wave arrays[END_REF], the proposed antenna is fabricated only on two layers and employs multiple annular slots to provide dual-band dual-CP operation in Ka-band with low profile, low return loss, very low axial ratio and narrow beam coverage of ±30°. L. Roy is with the Department of Electronics, Carleton University, Ottawa, ON K1S5B6, Canada (e-mail: lroy@doe.carleton.ca).

It uses sequential rotation to obtain a broad impedance and axial-ratio bandwidth.

In Section II, the measurement of miniaturized single elements using annular slots is presented and the dual-band dual-CP principle is explained. Section III summarizes the performance of experimental 2×2 subarrays with dual-band dual-CP capability. Finally, in Section IV, conclusions are drawn.

II. EXPERIMENTAL VALIDATION OF MINIATURIZED ELEMENTS FOR DUAL-BAND DUAL-CP

Printed slot antennas have attractive features compared to patch antennas: larger bandwidth, lower profile and lower conductor loss [START_REF] Garg | Microstrip Antenna Design Handbook[END_REF]. Moreover, they can easily produce two different operating frequency bands and CP radiation [START_REF] Ren | Design of dual-band circularly polarized annular slot antenna for wireless local area network applications[END_REF][START_REF] Cai | Dual-frequency microstrip antenna with dual circular polarization[END_REF][START_REF] Bao | Dual-Frequency dual-sense circularly polarized slot antenna fed by microstrip line[END_REF][START_REF] Ding | Dual-band circularly polarized dual-slot antenna with a dielectric cover[END_REF][START_REF] Bao | Monofilar spiral slot antenna for dualfrequency dual-sense circular polarization[END_REF][START_REF] Row | Design of aperture-coupled annular-ring microstrip antennas for circular polarization[END_REF]. The proposed radiating element (Fig. 1) then consists of two miniaturized annular ring slots. The larger slot operates at 20.5 GHz and is LH circularly polarized while the smaller one operates at 30.5 GHz and is RH circularly polarized. Each slot is fed by 2 microstrip lines with phase quadrature in order to produce CP. CP could also be obtained from a single feedline by adding stubs to the annular ring slot (as in [START_REF] Kim | Reconfigurable Annular Ring Slot Antenna with Circular Polarization Diversity[END_REF]) or by using a L-shape microstrip line (as in [START_REF] Mu | An annular-ring-slot antenna for circular polarization[END_REF]). However, it is well-known [START_REF] Garg | Microstrip Antenna Design Handbook[END_REF]- [START_REF] Kou | Frequency and pattern reconfigurable annular slot antenna with two feeding ports[END_REF] that 2 feedlines improve the overall CP bandwidth. The chosen substrate is the moderately high permittivity Rogers TMM10i substrate (h=0.38mm, ε r =9.8) supporting the metallic ground plane (including the annular ring slots) and the feedlines. It is glued with a 3M bonding film on a 4mm-thick low-permittivity foam layer. The height h between the radiating elements and the additional ground plane is optimized to 4mm (cf. Fig. 1).

Both elements can be accommodated in a 7×7 mm 2 lattice. This corresponds to an array with 0.5λ 0 and 0.7λ 0 inter-element spacing at 20 and 30 GHz respectively, thus allowing beam scanning up to ±60° and ±30° for downlink and uplink. GHz bands for the downlink and uplink antenna respectively. The shift (uplink band) of the measured reflection coefficient towards the lower frequency are attributed to fabrication imperfections and perturbations brought by the connector. In Fig. 4, the measured gain in each band agrees well with the simulated value (measured maximum gain in the broadside direction G max is 1.2 dB and 1 dB for the downlink and uplink antennas, respectively). Note that an additional ground plane is required to suppress backward radiations and to obtain a higher gain. The measured gain of the Tx element is noisy, because of the small dynamic range of the test setup and also due to the perturbation by the connector. Fig. 5 shows the measured and simulated axial ratios. The measured results are not as good as the simulated performance: AR simu =2.5dB and AR meas =4.1dB at 20.5 GHz and AR simu =1.4dB and AR meas =3.2dB at 30.5 GHz, in the broadside direction.

In the next section, we demonstrate how this initial performance may be improved at the array level using sequential rotation.

III. FOUR-ELEMENT SUBARRAY MEASUREMENT

Two 2×2 subarrays are now studied. These are formed by four annular ring slots arranged sequentially in orientation and in phase to achieve radiation symmetry, to cancel undesirable higher-order modes and to obtain purer polarization [START_REF] Jazi | Design and implementation of circularly polarized microstrip antenna array using a new serial feed sequentially rotated technique[END_REF]. For the sake of simplicity, 2 different subarrays will be considered as shown in Fig. 6. In the first one, four slots working at 20 GHz are fed using sequential rotation and in the second one, the same process is applied to four slots working at 30 GHz. Non excited elements working in the other band are also included to account for possible mutual coupling. Both subarrays can thus be seen as specific sets of elements extracted from a larger interlaced array (as shown in Fig. 6). To demonstrate the proposed concept, a prototype of each subarray has been fabricated and tested. Their pictures are shown in Figs. 7 and8, respectively for the 20 GHz-band and 30 GHz-band. The coupling between ports is lower than -19dB across the entire bandwidth. Moreover, a very good reflection coefficient is obtained for each port at the central frequency (S 11 /S 22 /S 33 /S 44 ≤-16dB in the two frequency-bands). The large difference between simulation and measurement is mainly due to the fabrication and assembly imperfections and the effect of the connector. The experimental radiation pattern in circular polarization is plotted in Fig. 11 for both frequency bands against elevation angle θ. Note that during the measurements, an absorbing layer is placed behind each subarray. For completeness, the axial ratio of both subarrays is shown in Fig. 12 against frequency for the broadside direction.

In Fig. 11, the measured gain in each band agrees well with the simulated result (measured maximum gain in the broadside direction G max is 4 dB and 5 dB for the 20 GHz-and 30 GHzsubarrays respectively). The improvement of the axial ratio is clearly demonstrated in the broadside direction: the axial ratio bandwidths of 3 dB are better than 3GHz for both frequency bands. Beam coverage of ±30° can thus be obtained for the downlink and uplink frequencies.

IV. CONCLUSION

A novel Ka-band dual-band dual-CP antenna array is presented in this letter. Two interlaced arrays each using an appropriate sequential rotation have been applied to improve the polarization purity and to generate a LHCP radiation on the downlink frequencies and a RHCP radiation of the uplink frequencies. Both measured and simulated results verified the validity of the proposed concept. The results show that the proposed 2×2 subarrays have good radiation patterns and high polarization purity (axial ratio at broadside less than 1.5 dB and 0.5 dB respectively) over a frequency band of 19-22 GHz and 29-32 GHz.

  This work was supported by the Direction Générale de l'Armement (DGA) of France and the Natural Sciences and Engineering Research Council (NSERC) of Canada. S. Mener and R. Gillard are with the Institute of Electronics and Telecommunications of Rennes, INSA, UMR CNRS 6164, 35708 Rennes (email: simon.mener@insa-rennes.fr).

  Fig. 1. Proposed dual-band dual-CP antenna with miniaturized annular ring slots in Ka-band. (a) Top view: a=7mm, R1=1.25mm, R2=0.95mm, R3=0.9mm, R4=0.6mm (b) 3D view: h=4mm. A Dual-Band Dual-Circular Polarization Antenna for Ka-Band Satellite Communications Simon Mener, Raphael Gillard, Langis Roy B Each slot element has first been optimized individually with the HFSS commercial software package. Prototypes have then been fabricated and tested. They are shown in Fig. 2. The experimental results of reflection coefficient, gain, and axial ratio against frequency or elevation angle θ are shown in Fig. 3-5, respectively. Note that during the measurements, an absorbing layer is placed behind each antenna to suppress backward radiations and associated disturbances.

Fig. 2 .Fig. 3 .

 23 Pictures of the fabricated miniaturized annular ring slot antennas in Kaband. (a) Back view of the 20 GHz-band prototype. (b) Front view of the 20 GHz-band prototype. (c) Back view of the 30 GHz-band prototype. (d) Front view of the 30 GHz-band prototype. Simulated and measured reflection coefficient of the miniaturized annular slot antennas. (a) 20 GHz-band. (b) 30 GHz-band.

Fig. 4 .

 4 Simulated and measured gain of the miniaturized annular slot antennas. (a) 20.5 GHz in the x-z plane. (b) 30.5 GHz in the x-z plane.

Fig. 5 .

 5 Simulated and measured axial ratio of the miniaturized annular slot antennas. (a) 20.5 GHz in the x-z plane. (b) 30.5 GHz in the x-z plane.In Fig.3, both the measured and simulated -10dB reflection coefficient cover the[19.5-21.5] GHz and[29][30][31]

Fig. 6 .

 6 Fig. 6. Dual-band/ Dual-CP concept: two interlaced arrays with appropriate sequential rotation in Ka-band.

Fig. 7 .Fig. 8 .

 78 Pictures of the fabricated subarray composed of four elements in sequential rotation in the 20 GHz-band. (a) Back view. (b) Front view. Pictures of the fabricated subarray composed of four elements in sequential rotation in the 30 GHz-band. (a) Back view. (b) Front view. The simulated and experimental S-parameter results are presented in Figs 9 and 10 for the 20GHz-and 30 GHzsubarrays respectively.

Fig. 9 .

 9 Simulated and measured reflection coefficient of the 2×2 subarray against frequency. (a) 20 GHz-band. (b) 30 GHz-band. (a) (b) Fig. 10. Simulated and measured transmission coefficient of the 2×2 subarray against frequency. (a) 20 GHz-band. (b) 30 GHz-band.

Fig. 11 .

 11 Simulated and measured gain of the 2×2 subarray. (a) 20.5 GHz in the x-z plane. (b) 30.5 GHz in the x-z plane.

Fig. 12 .

 12 Simulated and measured axial ratios for a broadside direction (θ=0°) against frequency. (a) 20 GHz-band. (b) 30 GHz-band.


Simon Mener
email: simon.mener@insa-rennes.fr

Raphaël Gillard

Langis Roy
email: lroy@doe.carleton.ca

A Novel Ka-band GaN Phase Shifter for SATCOM Phased-Array Antennas

Keywords: Gallium-nitride (GaN), phase shifter, Ka-band

A Ka-band phase shifter is a key component of phased array antennas for SatCom applications. By using GaN technology, large power handling capability and high frequency operation can be achieved. In this paper, a HEMT GaN switch using a metal gate of 0.15μm is designed at 30 GHz for signal levels up to 37 dBm. From this efficient GaN switch, a compact and innovative monolithic phase-shifter is proposed for Ka-band operation with a 2-bit phase resolution and attractive insertion loss, isolation and bandwidth performance.

I. INTRODUCTION

Because satellite communication (SatCom) systems having large capacities are in demand, Ka-band phased array systems are being investigated. This higher operational frequency provides at the same time high data rate links and miniaturized antenna systems [START_REF] Marcellini | Future developments trend for Ku and Ka antenna for satcom on the move[END_REF]. Traditionally, beam scanning is achieved through mechanical approaches [START_REF] Kot | Antenna concepts for Kaband SATCOM on the move[END_REF][START_REF] Kot | Compact Shaped Dual-Reflector System For Military Ka-band SATCOM on the move[END_REF], which are bulky, slow, and expensive. For this reason, an electrically controlled phase array antenna is preferred. A phase shifter behind each radiating element is desired to sweep the direction of the beam. Moreover, a compact and accurate 2-bit phase-shifter is required to allow an appropriate beam scanning and reduce the dimensions of the phase control circuit in T/R modules (which generally occupies a large area in SatCom systems).

In addition, research into wide-bandgap semiconductors has progressed significantly over the last few years. GaN based semiconductors have demonstrated excellent performance in comparison with the other wide band-gap semiconductor technologies [START_REF] Runton | History of GaN: highpower ,RF gallium nitride (GaN) from infancy to manufacturable process and beyond[END_REF]. Large power handling capability and high frequency operation are achieved with good efficiency and linearity. As a result, reconfigurable RF circuits with high power/frequency and excellent linearity are now realizable, especially for Ka-band SatCom systems.

In this paper, a compact Ka-band (30-32 GHz) 2-bit phase shifter employing GaN switches is proposed for a new generation of Ka-band phased array systems for satellite communication. A Ka-band GaN HEMT switch has been designed (and laid out for fabrication at NRC) to be used in a phase-shifter system. The general characteristics, dimensions and performances for On-and Off-Mode GaN switch configurations are detailed in Table I and Fig. 2. N is the number of gate fingers, W and L are respectively the width and the length of gate finger. After optimization (insertion loss and isolation at 30 GHz), the optimal width of the gate W is found to be 120μm. Note that the final dimensions of this innovative and efficient GaN switch in Ka-band are 220μm×450μm.

II. DESIGN OF KA-BAND GAN SWITCH

Table I. GaN switch details and simulated performances in Ka-band To take into account the switch performances in the forthcoming phase-shifters, modeling has been conducted to extract the value of the parasitic element for each state (Onand Off-Mode). As described in Fig. 3, the physical equivalent circuit can be represented by a simple resistance for the On-Mode (optimal resistance R on =12 Ω) and by a capacitance for the Off-Mode (optimal capacitance C off =12.3fF). Fig. 3 shows the performances of the proposed GaN switch in On-Mode (Fig. 3a: insertion loss against frequency) and in Off-Mode (Fig. 3b: isolation against frequency) in comparison to the proposed simple equivalent circuit. Note that the insertion loss is around 1.2 dB and the isolation is lower than 13 dB for all frequencies respectively for the On-and Off-Mode. The component values have been correctly optimized: the curves are very similar over the entire bandwidth. Note that the 1-dB compression point is a little over 37 dBm. Generally, an active phased array antenna is required to operate over a wide scanning frequency bandwidth of more than 1GHz. The proposed approach is to associate a phase shifter for each element with an efficient true time delay. The true time delay performance is obtained by controlling the electrical path-length of each element. The proposed Ka-band compact phase-shifter is shown in Fig. 4. The length of each electrical meander microstrip path has been optimized to obtain four phase-states separated by 90 degrees and to reduce the overall size. Note that the GaN MMIC corresponds to a substrate with permittivity ε r of 10 and a thickness of 75μm. GaN switches are needed to select an individual path. So, to achieve four 90°-spaced phases (namely a 2-bit phase resolution), two GaN switches in On-Mode and six GaN switches in Off-Mode are required for each phase-state. The required gate bias circuitry to control each switch is not shown, but could be easily added without affecting the size or performance of the phase shifter. The frequency response computed in transmission with HFSS is represented in Fig. 5 both in amplitude (Fig. 5a) and in phase (Fig. 5b). In all simulations, for each phase-state (4 simulations/4 states: 1 st : switches n°1 in On-mode and the others in Off-Mode, 2 nd : switches n°2 in On-Mode etc.), the parasitic elements of GaN switches are included: resistance of 12 Ω for On-Mode and capacitance of 12.3fF in Off-Mode. Regardless of the active phase-state, the transmission coefficient is lower than 3 dB between 30 and 32 GHz. The transmission phase response is plotted in Fig. 5b between 30 and 32 GHz; four 90°-spaced phase configurations are obtained with the same frequency dispersion. The corresponding phase resolution is around 1.95bits. To conclude, Table II compares the proposed phase shifter with those currently reported in the literature at Ka-band. This comparison demonstrates the value of this innovative phase-shifter. It is compact and efficient (insertion loss, bandwidth, phase resolution). Moreover, it includes GaN switches to handle high power at Ka-band.

IV. CONCLUSION

An innovative phase-shifter composed of GaN switches in Ka-band is proposed in this paper. A HEMT GaN switch using a metal gate of 0.15μm (NRC GaN 150 process) has been designed and simulated in collaboration with the NRC. The simulated insertion loss is 1.2 dB and the isolation is lower than 13 dB at 30 GHz. Subsequently, a compact and efficient phase-shifter has been optimized and designed for Ka-band operation taking into account the parasitic elements of the GaN switches. In comparison to the other Ka-band phase-shifters, performances are very attractive: the proposed phase-shifter is compact and efficient in terms of insertion loss, isolation, bandwidth and phase resolution.

Fig. 1

 1 Fig. 1 shows a simplified cross-section of the GaN HEMT process, which has been developed by the National Research Council Canada (NRC). The NRC GaN technology is an aluminum -gallium -nitride / gallium -nitride (AlGaN/GaN)

Fig. 1

 1 Fig. 1 Schematic cross-section of AlGaN/GaN NRC HEMT

Fig. 2 .

 2 Proposed HEMT GaN switch with test structure. (a) 3D general view of the layout. (b) Top view of the layout with dimensions.

Fig. 3 .

 3 Performances of the proposed GaN HEMT switch in comparison with a simple equivalent circuit. (a) ON-MODE: Insertion loss against frequency. (b) OFF-MODE: Isolation against frequency.III. DESIGN OF A KA-BAND GAN PHASE SHIFTER

Fig. 4 .

 4 Proposed Ka-band phase shifter using HEMT GaN switches. (a) Top view with general dimensions. (b). Top view of the layout with test structure (a) (b) Fig. 5. Simulated performances of the proposed phase-shifter for the four phase states. (a) Magnitude of the transmission coefficient S21. (b) Phase of the transmission coefficient S21. Prototypes of a nearly identical passive phase shifter (but realized on Duroid 5880 without GaN devices) have been fabricated and gave excellent measured results [DGA report], suggesting that experimental performance of the proposed GaN MMIC circuits will closely match simulations. Chip fabrication is planned in 2016.

Table II .

 II Ka-band phase-shifter comparison

	[Reference]				Phase
	Simulated or measured	Operating frequency (GHz)	Insertion loss (dB)	Isolation (dB)	variation Continuous or	Size (mm)
	results				Discrete
	[4] Measured	DC-33	8	10	360° Continuous	7.61×1. 2
	[5] Measured	30	5,8	12	157° Continuous	4.2×0.9
	[7] Simulated	22-26	≤10	10	25,4° Continuous	1.45×0. 98
	[8] Measured	15-35	≤10	10	360° Discrete	0.52×0. 37
	[9] Measured	27-32	5 ± 0,8	10	360° Continuous	2×1.2
	[10] Simulated	30-30,5	3,6 ± 1,8	10	315° Continuous	20×20
	[11] Measured	30	5,8	12	157° Continuous	?
	[12] Measured	20-25,75	1	20	330° Discrete	0.8×2.3
	This work Simulated	30-32	2,5

13 360° (2bits) Discrete 2.7×3.6