Antenna De-Embedding of Radio Propagation Channel With

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IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 1 Antenna De-embedding of Radio Propagation Channel with Truncated Modes in the Spherical Vector Wave Domain Yang Miao, Student Member, IEEE, Katsuyuki Haneda, Member, IEEE, Minseok Kim, Member, IEEE, and Jun-ichi Takada, Senior Member, IEEE

Abstract—This paper proposes a novel approach to extracting of antennas the system should be re-designed. Hence, it helps the narrowband propagation channel from the communication to separately deal with the impact of the antennas from the link by de-embedding the impact of the antennas. In the proposed radio propagation channel for more efficient system design. approach, the mode-to-mode mapping matrix M which is the The conventional plane wave channel modeling is based expression of the propagation channel in the spherical vector wave on the double-directional characterization of the propagation domain, is estimated by applying pseudo-inverse computation to the channel transfer functions with dedicated spherical arrays. channel. Its focus is on estimation and parameterization of The estimated M is truncated and only the dominant modes well-known parameters, e.g., the angle of departure (AoD), within the spatial bandwidth of the fields radiated from the finite angle of arrival (AoA), and polarimetric complex gain of each volume of the spherical array are considered. Two types of spher- multipath component [1]–[3]. ical array are investigated: an ideal array using tangential dipoles Alternatively, spherical vector wave channel modeling, and a virtual array using dielectric resonator antenna (DRA). which expands the radio propagation channel into a product of The ideal array is used for parameter investigation including the three terms all expressed by spherical vector wave expansion array radius and the spacing with regard to the size of M and coefficients, has been proposed in [4]. The antennas are mod- the condition number of the excitation coefficient matrix, while eled by the coefficients of the spherical vector wave expansion the virtual DRA array is proposed as a practical implementation of the ideal array. The accuracy of the proposed approach has [5]. The behavior of the propagation channel is modeled by been validated numerically. The uncertainties of spherical array a mode-to-mode mapping matrix M, which describes the in practice, such as the influence of non-ideally embedded array linear relationship between the radiated and the impinging elements, cables and fixtures, are considered in the validation. spherical vector waves. This alternative description of the radio Moreover, the channel transfer functions reproduced by the de- propagation channel formulates a clear separation between embedded M are analyzed given different target antennas at link the antennas and the propagation channel. This has a few ends. The gain and phase discrepancies as well as the antenna implications of great practical importance. First, it is well- correlations of the reproduced channel transfer function are known from antenna near-field measurement theory that an compared with the generated reference. antenna’s radiated field can be expanded into a finite number Keywords—mode-to-mode mapping matrix, antenna de- of spherical vector waves, hence the propagation channel can embedding, truncated modes, ideal spherical array, virtual be modeled by a finite number of modes. In the case of spherical DRA array electrically small antennas, this may drastically reduce the number of modeling parameters. Second, the spherical vector wave expansion does not constrain the type of electromag- I.INTRODUCTION netic wave impinging at the receive antenna. There are no Multiple-Input Multiple-Output (MIMO) radio propa- restrictions on the physical mechanism that create the waves A gation channel comprises the propagation channel and impinging the receive antenna. Hence, arbitrary antennas and the antenna arrays at both ends of the communication link. propagation channels can be studied with the spherical vector While the antennas are designable components, the propaga- wave expansion approach. tion channel is usually determined by the physical environment As a result, new insights about antenna–channel interaction surrounding the antennas. The successful deployment of a have been gained. In [6], [7], the mean effective gain (MEG) MIMO wireless system depends, among many other things, is formulated, which yields maximum MEG condition as well on the performance of the employed antennas. For the change as limitation on MEG and radiation quality factor Q of an antenna. In [8], the cross-correlation between two antenna Y. Miao, J. Takada are with the Department of International De- velopment Engineering, Tokyo Institute of Technology, P.O.Box S6- branches is analyzed. In [9], [10], the spatial degree of freedom 4, 2-12-1, O-okayama, Meguro-ku, Tokyo, 152-8552 Japan (e-mail: is obtained, which is further applied to compare the spatial [email protected]). multiplexing and the beamforming in [11]. K. Haneda is with the Department of Radio Science and Engineering, In the spherical vector wave channel modeling, M repre- School of Science and Technology, Aalto University. M. Kim is with the Department of Electrical and Electronic Engineering, sents the compact version of all the signal behaviors in the School of Engineering, Niigata University. propagation environment, and the modeling as well as the Manuscript received month.day, year; revised month, year. estimation of M are necessary. In [4], it is shown that the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 2 entries of M under Rayleigh fading are Gaussian variates different antennas in the same propagation environment whose statistics are functions of power angular spectrum. In are compared in a straightforward manner. [12], the clustering impact on the statistics of M is numerically The remainder of this paper is organized as follows. In studied. In [10], M is estimated by converting the realizations Section II we review the plane wave representation and the in the plane wave channel modeling, where the parameters spherical vector wave representation of the MIMO channel. In are obtained by applying SAGE algorithm to the measured Section III we propose the antenna de-embedding approach by radio channel transfer functions. In [13], the computation of using the ideal spherical array and the virtual spherical DRA M using the finite-difference time-domain (FDTD) method array, and validate the approach by numerical simulations con- is proposed, where the single-mode spherical wave source is sidering the uncertainties in practice. In Section IV we examine generated at the transmit antenna side by a cubical dipole array the reproducibility of channel transfer function synthesized by referencing [9], [14]. In [15], the estimation of spherical vector the de-embedded M given different target antennas at link wave coefficients from channel measurement by using 3-D ends. In Section V we conclude this paper. positioner is discussed, where the receive antenna is virtually positioned inside or on the surface of a cube. II.MIMOCHANNEL REPRESENTATION As can be seen from above, on one hand, the estimation of M is typically conducted with the help of plane wave channel In this section we review the MIMO channel representations model parameters [4], [10], [12]. However, the parameter in both the plane wave and the spherical vector wave domains. jwt estimation in the plane wave domain often discards the diffuse Time dependence in this paper is e . scattering, and the M generated by using these plane wave parameters cannot take full advantage of the spherical vector A. Representation in Plane Wave Domain wave channel modeling. One the other hand, the research In the plane wave domain, MIMO channel transfer matrix activities of estimating M without using plane wave channel × H ∈ CNr Nt [16] is represented by model parameters are limited [13], [15]. But in [13], although ∫ ∫ spherical wave sources are used, the proposed approach is H = A (κˆ)α(κˆ, kˆ)AT (kˆ)dkˆdκˆ (1) only optimized for the scheme of numerical computation, r t Ωr Ωt but not applicable for the practical measurement because the real antenna configuration is not considered. In addition, the where Nt and Nr denote the numbers of the transmit and measurement positions of receive antenna in [15] are under receive antennas, respectively. Ωt and Ωr are the channel random trials without configuration optimization. solid angles subtended by the scatterers as viewed from the ˆ Therefore, in this paper, we estimate the mode-to-mode transmit and receive antennas, respectively. k = [θt, ϕt] and ˆ mapping matrix M with more practical considerations, so κˆ = [θr, ϕr], where k and κˆ are unit vectors containing the ˆ that the proposed approach is instructive for the practical k ∈ Ωt and κˆ ∈ Ωr respectively. θt, θr are elevation angles, ˆ 2×2 measurement and also can be examined in terms of channel and ϕt and ϕr are azimuth angles. α(κˆ, k) ∈ C is the modeling accuracy. The major contributions are threefold: matrix containing the polarimetric complex gains of a plane i This paper proposes an ideal spherical array with tan- wave and is [ ] α α gential dipoles for estimating M from Hsph which is α = VV VH . (2) the radio channel transfer function with the spherical αHV αHH arrays at link ends. We introduce new criteria for At and Ar defined by array configuration, where the array radius and the [ ] × ˆ ˆ ˆ Nt 2 spacing are determined according to the dimension At(k) = at,V(k) at,H(k) ∈ C (3) of M as well as the pseudo-inverse condition of the × A (κˆ) = [ a (κˆ) a (κˆ) ] ∈ CNr 2 excitation coefficient matrix. The proposed approach r r,V r,H with the ideal spherical array is validated by numerical are the array response matrices of the transmit and the receive simulation under various propagation environments. antennas, respectively. (·)T denotes vector/matrix transpose. ii A virtual spherical array is introduced to estimate The propagation channel representation in the plane wave M as a practical implementation. The array element domain can be described by the statistical distributions of kˆ, is the dielectric resonator antenna (DRA), and the κˆ and α(κˆ, kˆ). For the statistical description of α(κˆ, kˆ), the array configuration satisfies the configuration of the cross-polarization power ratio (XPR) and the co-polarization ideal spherical array. We analyze how de-embedding power ratio (CPR) are used, which are defined in this study as the virtual array effects the accuracy of our results, follows: ∫ ∫ ( ) where different uncertainty levels are added to the 2 2 |αVV| + |αHH| dΩtdΩr excitation/weighting coefficients of spherical array to XPR = ∫ ∫ (4) | |2 | |2 represent the practical issues such as the non-ideally ∫ ∫ ( αVH + αHV ) dΩtdΩr 2 embedded antenna element, the impact of cables and |αVV| dΩtdΩr fixtures, and so on. CPR = ∫ ∫ . |α |2dΩ dΩ iii The validations in terms of the channel transfer func- HH t r tion reproduced by the de-embedded M given target Although general channel model descriptions are consid- antennas are conducted, where the performances of ered, some parameters need to be defined specially for the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 3

Line-of-Sight (LOS) environment. In particular, the ratio of C. Conversion Formula between Plane Wave and Spherical the power of the dominant path (denoted as ”dm”) to the mean Vector Wave Domains power of the fading paths (denoted as ”fd”): Assume the array responses in (1) are the antenna |αdm |2 + |αdm |2 + |αdm |2 + |αdm |2 gains. Conventionally, by expanding the array responses into γ = VV VH HV HH (5) weighted spherical vector waves [5, p. 55] and using reci- E{|αfd |2 + |αfd |2 + |αfd |2 + |αfd |2} VV VH HV HH procity, the entry of the mode-to-mode mapping matrix M in where E{·} denotes the average over the ensemble. (6) can be obtained by using the channel model parameters in plane wave domain [4]: In this study, it is assumed that the correlations at transmit ∫ ∫ antennas and the correlations at receive antennas are inde- − − H ′ − n+υ+4 s σ · ˆ pendent and separable. Hence, no matter indoor or outdoor Mjj = ( j) Ysmn (κˆ) α(κˆ, k) (8) measurements, we assume the main scatterings appear close Ωr Ωt ˆ ˆ to the antenna arrays at both link ends [17]. · Yσµυ(k) dkdκˆ where the prefix coefficient (−j)n+υ+4−s−σ corresponds to ˆ B. Representation in Spherical Vector Wave Domain the far-field Hankel functions. Ysmn(κˆ) and Yσµυ(k) are the In the spherical vector wave domain, the narrowband chan- spherical vector harmonics: [ ] nel transfer function of a MIMO channel can be defined as a ˆ T 2×1 Yσµυ(k) = y (kˆ) · θˆ y (kˆ) · ϕˆ ∈ C (9) linear combination of the physical modes of the antennas at [ σµυ σµυ,H ] ˆ ˆ T ∈ C2×1 both ends and the propagation channel [4]: Ysmn(κˆ) = ysmn(κˆ) · θ ysmn(κˆ) · ϕ H = RMT (6) Please refer to Appendix A for more details. Note that the ′ × × × single indices j and j are convertible with the triple indices where M ∈ CJr Jt , T ∈ CJt Nt , and R ∈ CNr Jr are {σµυ} and {smn} respectively [5, p. 15]. the mode-to-mode mapping matrix, the transmitter’s excitation coefficient matrix, and the receiver’s weighting coefficient III.PROPOSED ANTENNA DE-EMBEDDING APPROACH matrix, respectively. Jt and Jr are the numbers of the spherical vector wave modes of the transmit and receive antennas, This section presents an antenna de-embedding approach to respectively. estimate the mode-to-mode mapping matrix from the channel Jt and Jr are determined by the the minimum radii of the transfer functions of multiple-antenna channels, which we transmit and the receive antennas, which are denoted by rt and usually measure in channel sounding. The proposed approach rr, respectively [5] as follows can be applied to data collected with the specially designed spherical array. We derive the conditions required to achieve Jt = 2(⌊krt⌋ + n0) {(⌊krt⌋ + n0) + 2} (7) this purpose. Two types of the arrays are considered: an Jr = 2(⌊krr⌋ + n0) {(⌊krr⌋ + n0) + 2} ideal array using tangential dipoles for investigating general array configuration and a virtual DRA array for practical 2π ⌊·⌋ where λ, k = λ , and are the wave length, the wave consideration. number, and the floor function, respectively. n0 is determined based on the accuracy of the spherical vector wave expansion, A. Proposed Approach specifically the number of evanescent modes to be considered. Note that the evanescent modes, in this paper, refer to the The maximum number of modes employed in the antenna spherical vector wave modes outside the spatial bandwidth of de-embedding is denoted by Jt,sph for the transmit antenna ≤ the limited size of the source. While n = 10 is often applied side and Jr,sph for the receive antenna side. Hence, Jt Jt,sph 0 ≤ [5], other criteria have been also proposed and utilized [14], and Jr Jr,sph should be satisfied. The spherical vector [18], [19]. wave coefficients of the transmit and receive antennas used for ′ ′ Jt,sph×Nt,sph antenna de-embedding are denoted by Tsph ∈ C The (j , ι )-th element of T represents the coefficient of × ′ ′ ∈ CNr,sph Jr,sph the j -th spherical vector wave mode for the ι -th transmit and Rsph , respectively. The estimated mode- × ˆ Jr,sph Jt,sph antenna. Similarly, the (ι, j)-th element of R represents the to-mode mapping matrix is denoted by Msph ∈ C . coefficient of the j-th spherical vector wave mode for the ι- Equation (6) is then satisfied by Tsph, Rsph, and Mˆ sph. † th receive antenna. T can be calculated from the antenna’s Multiplying the Moore-Penrose pseudo inverse (·) of Tsph radiation pattern by the inner product method [5, p. 96] or and Rsph at both sides of (6), the following equation can be the least square solution method [18]. R can be calculated by obtained: ˆ † † reciprocity [5, p. 36]. Msph = RsphHsphTsph. (10) M describes the linear relationship between the outgoing 1 ˆ ˆ and the incoming spherical vector wave modes. The entry M is obtained by truncating the estimated Msph: Mjj′ represents the transfer function of the radio propagation ′ Mˆ ′ = Mˆ ′ (11) between the j -th transmit and j-th receive mode. In this paper, (j,j ) sph(j,j ) the estimation of M, from the channel transfer function matrix 1Note that the ”pinv” function in Matlab is used in practice for this Moore- with specially designed antenna array, by solving the inverse Penrose pseudo inverse. It is based on singular value decomposition (SVD) computation problem of (6) is called antenna de-embedding. and any singular values less than a tolerance are treated as zero. IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 4

-pol e & m dipoles 2) spacing: The spacings for the transmit and receive spher- -pol e & m dipoles ical arrays are denoted by ∆dt and ∆dr, respectively. With ∆dt, the number of array elements Nt,sph, which is necessary to examine condition (i), can be calculated by:     z πg G∑−1 2πr sin   t,sph G   spacing N = 4    + 2 (14) t,sph ∆d (approximately g=1 t equidistant) [ ] πrt,sph G = (15) ∆dt y where the summation of an extra 2 corresponds to the locations at the north and the south poles in the array sphere, and [ · ] denotes the nearest integer function. This calculation is x based on the design process of the approximately equidistant Fig. 1. The ideal spherical array for antenna de-embedding spherical grid, where firstly the elevation angles are evenly divided and then the azimuthal angles at each elevation level are divided. When ∆dt becomes smaller, Nt,sph becomes ′ where j = 1 ... Jt and j = 1 ... Jr. In order to obtain accurate larger, and the condition number decreases to approach 1. ˆ estimates of Msph through (10), Tsph and Rsph should satisfy Hence, smaller ∆dt is more advantageous for satisfying the the following two conditions: condition (i) and (ii). However, a large Nt,sph results in a high i The number of array elements should be larger than the cost of computation and measurement. Therefore, to consider number of expanded spherical vector wave modes, that the trade-off, ∆dt is determined as the maximum spacing is, Nt,sph > Jt,sph and Nr,sph > Jr,sph. In other words, which satisfies the condition (i) and (ii). This design can be the over-determined array configurations are required for made by experimentally setting ∆dt between 0.4λ and 0.5λ, both the transmit and receive antennas. then the largest possible ∆dt can be decided by checking the ii The condition number [20] of Tsph and Rsph should be obtained Nt,sph as well as the condition number of Tsph. The close to one to decrease the pseudo-inverse error of the same criteria is valid for Nr,sph and ∆dr by replacing rt,sph linear inverse computation problem. with rr,sph.

B. An Ideal Spherical Array C. A Virtual Spherical DRA Array We consider an ideal spherical array, as is shown in Fig. 1. At each point of the approximately equidistant spherical In practice, the compact MIMO DRA proposed in [21] can grid, θ− and ϕ− polarized electric and magnetic incremental be used to form a virtual spherical array satisfying the proposed dipoles are located. Hence at each point of the spherical grid configuration. As is shown in Fig. 2 (a), the DRA has 3 ports there are four elements. We assume all the elements are excited with orthogonal polarizations. The centered dielectric cube, the dielectric constant and loss tangent of which are 21 and equally, and each element has one port for the input signal. − The configuration parameters of the spherical array are the 1.35 × 10 4, is with side length of 18 mm. The antenna array radius and spacing, which can be designed specially to elements are painted with silver on the dielectric material, satisfy the condition (i) and (ii) in Subsection A. where port 1, 2 are 4 mm wide and 7 mm tall, and port 3 1) radius: The radii for the transmit and receive spherical is a 2 mm diameter cylinder with 10.5 mm tall. The DRA’s arrays are denoted as rt,sph and rr,sph, respectively. As is radiation pattern is simulated by using CST microwave studio. shown in [5], rt,sph determines the maximum modes that Hence, the expanded spherical vector wave coefficients can can be excited. Modes with order less than or equal to be calculated. As is shown in Fig. 2 (b)-(d), the power is 2⌊krt,sph⌋(⌊krt,sph⌋ + 2) are dominant modes, and the modes concentrated at the first 6 modes, indicating that the DRA with order larger than it are evanescent modes which can not radiates the mixed modes of dipoles. Port 3 radiates the 4- be excited fully. For satisfying the condition (ii), Jt,sph should th mode same as a vertical electric dipole, port 1 (or port not exceed the number of dominant modes: 2) radiates the 1-st and 5-th modes, the same as a horizontal magnetic dipole, as well as the 2-nd and 6-th modes, the same J ≤ 2⌊kr ⌋(⌊kr ⌋ + 2). (12) t,sph t,sph t,sph as a horizontal electric dipole [5, p. 39]. Hence, ⌈√ ⌉ By rotating and translating the DRA at designed grid lo- J cations and keeping the DRA ground tangential to the sphere t,sph + 1 − 1 2 surface, as illustrated in Fig. 3, virtually the spherical vector rt,sph ≥ (13) wave modes of the ideal spherical array in Subsection III-B k can be achieved. For this virtual spherical DRA array, there are where ⌈·⌉ denote the ceiling function. For rr,sph, same equation 3 ports at each location. To calculate the spherical vector wave as (13) is used by replacing Jt,sph with Jr,sph. coefficients Tsph,DRA at each location on sphere, we translate IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 5

TABLE I. ARRAY CONFIGURATION EXAMPLESOF IDEAL SPHERICAL z ARRAY 1 Jt,sph rt,sph ∆dt Nt,sph Condition 0.8 Jr,sph rr,sph ∆dr Nr,sph number AC 1: 30 0.5λ 0.44λ 76 1.63 0.6 AC 2: 48 0.75λ 0.52λ 120 1.38 0.4 AC 3: 96 1λ 0.48λ 240 1.89

y Amplitude of 0.2

0 5 0 10 20 30 x Spherical Vector Wave Coefficient Spherical Vector Wave Mode Index AC 1

sph AC 2 (a) (b) 4 AC 3

3 AC 2: ∆ d <= 0.52λ 1 1 t AC 3: ∆ d <= 0.48λ 0.8 0.8 t AC 1: ∆ d <= 0.44λ 0.6 0.6 2 t 0.4 0.4 Condition number of T

0.2 Amplitude of 0.2 1 0.3 0.35 0.4 0.45 0.5 *λ 0 0 0 10 20 30 0 10 20 30 Tested values of spacing Spherical Vector Wave Coefficient Spherical Vector Wave Mode Index Spherical Vector Wave Coefficient Spherical Vector Wave Mode Index

Fig. 4. The condition number of Tsph with different tested values of spacing (c) (d)

Fig. 2. The DRA (a) and the amplitude of spherical vector wave coefficients array configuration (AC) of the ideal spherical array are listed. of DRA’s (b) port 1; (c) port 2; (d) port 3. The phases of the coefficients of According to (11), different AC which leads to different Jt,sph port 1 and port 2 are orthogonal. and Jr,sph, are designed for different range of Jt and Jr. If Jt and Jr are given, and if we assume Jt,sph = Jt and Jr,sph = Jr to maximumly use the available modes, the spherical array radii rt,sph and rr,sph can be determined by (13). In terms of the spacing, ∆dt and ∆dr are determined as the maximum value which ensures the condition (i) and (ii) in Subsection A. For this, the condition numbers with different tested values of spacing ∆d = ∆dt = ∆dr are plotted in Fig. 4. As can be observed from the figure, the condition numbers tend to be convergent when the tested spacing is smaller than a threshold. When the tested spacing is larger than the threshold, the condition number increases dramatically. This threshold is set as the spacing for the spherical array to minimize the computation or measurement cost. For the AC examples, the thresholds are 0.44λ for AC 1, 0.52λ for AC 2, and 0.48λ for AC 3, respectively. Hence, 0.44λ, 0.52λ, 0.48λ were found to be the appropriate spacings for AC 1, AC 2, AC 3, respectively. 2) Virtual Spherical DRA Array: As is shown in Table II, Fig. 3. Virtual spherical array with rotated and translated DRA, where the array configurations of the virtual spherical DRA array are (x, y, z) is the global coordinate and (x′, y′, z′) is the local coordinate determined by following the values of array size and spacing of the ideal spherical array. Fig. 5 (a) show the amplitudes of the spherical vector wave coefficients Tsph,DRA of the virtual and rotate the spherical vector wave functions in primed and spherical DRA array with AC 4, and (b) show the amplitudes unprimed coordinates, as in Appendices A2 and A3 of [5]. of the spherical vector wave coefficients Tsph of the ideal spherical array with AC 1. As can be observed, for both cases, D. Numerical Examples of Array Configuration all the dominant modes are excited by different array elements. 1) Ideal Spherical Array: The configuration of the spherical array for antenna de-embedding, i.e. rt,sph, rr,sph, ∆dt, ∆dr, E. Numerical Evaluation of De-embedding Accuracy can be designed according to the dimension of the desired The proposed antenna de-embedding approach is validated Mˆ , i.e. Jt, Jr. As is shown in Table I, three examples of by examining its accuracy. The simulation process is shown IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 6

TABLE II. ARRAY CONFIGURATION EXAMPLESOF VIRTUAL SPHERICAL DRAARRAY

Jt,sph rt,sph ∆dt Nt,sph Condition Jr,sph rr,sph ∆dr Nr,sph number AC 4: 30 0.5λ 0.44λ 57 3.64 AC 5: 48 0.75λ 0.52λ 90 2.36 AC 6: 96 1λ 0.48λ 180 4.68

(a)

Fig. 6. Simulation process for evaluating the accuracy of the proposed antenna de-embedding approach. The rounded rectangles indicate the initial settings, the rectangles indicate the simulation procedures, the dashed arrows indicate the corresponding data, and the solid arrows indicate the order of simulation.

Sight (NLOS) and LOS, as is shown in Table III, are consid- ˆ ered in our simulation. In Table III, k, κˆ, αVV, αVH, αHV, αHH are generated independently and separately according to statistical distributions. For kˆ, κˆ, both the uniform distribution on sphere (representing rich scattering environment around antennas) and the Gaussian distribution on sphere (representing more general scattering environment around antennas) are (b) considered. Note that the Gaussian distribution on sphere is equivalent to the von Mises-Fisher distribution [22], where Fig. 5. The amplitudes of the spherical vector wave coefﬁcients of: (a) the the variance of the former is inversely proportional to the virtual spherical DRA array with AC 4; (b) the ideal spherical array with AC concentration parameter of the latter. In terms of αVV, αVH, 1. Array element indexes from 1 to the end indicate the element locations αHV, αHH, the XPR and γ are determined according to a from the north pole to the south pole on sphere. polarized indoor MIMO channel measurement at 2.45 GHz in [23]. Another reason to choose MIMO channel measurement in in Fig. 6. As is shown in Fig. 6, the accuracies of antenna [23] is to support the frequency range of the DRA. Firstly the de-embedding with the ideal spherical array and the virtual de-embedding accuracies under ideal condition are evaluated, spherical DRA array are evaluated by comparing the de- then the analysis with practical consideration of uncertainties in spherical array are discussed. embedded Mˆ 1 and Mˆ 2 with the reference Mref , respectively. The reference Mref is generated by using the plane wave 1) Evaluation under Ideal Condition: We start with the channel model parameters according to the conversion formula numerical simulation under ideal condition without the con- (8). In (8), the discrete plane waves with super high resolution sideration of practical issues. Fig. 7 shows examples of how ˆ ˆ (density) are summed up to numerically approach the integral M1, M2, and Mref look like comparably under CM 4, where of continuous plane waves. Mˆ 1 is de-embedded by using the ideal spherical array with Four instances of channel model (CM), both Non-Line-of- AC 1 and Mˆ 2 is de-embedded by using virtual spherical DRA IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 7

TABLE III. CHANNEL MODEL (CM)PARAMETERSIN PLANE WAVE DOMAIN

CM 1 (NLOS) CM 2 (LOS) CM 3 (NLOS) CM 4 (LOS) kˆ Uniformly distributed von Mises-Fisher distributed with mean direction π π κˆ on sphere [θ, ϕ] = [ 2 , 4 ] and concentration parameter 10 αVV Complex Gaussian Complex Gaussian Complex Gaussian Complex Gaussian α distributed distributed fading part distributed distributed fading part VH ∼ ∼ α XPR= 8 dB + dominant path γ = 5 dB XPR= 8 dB + dominant path γ = 5 dB HV ∼ ∼ ∼ ∼ ∼ ∼ αHH CPR = 0 dB XPR= 16 dB, CPR= 0 dB CPR= 0 dB XPR= 16 dB, CPR= 0 dB array with AC 4. Both agreement and discrepancy are observed dB degrees by visual inspection. For quantitative evaluation, the amplitude 20 150 5 5 Mˆ Mˆ Mˆ 100 and phase discrepancies between estimated ( 1 or 2) 10 10 10 and Mref , ∆G and ∆P respectively, are deﬁned as follows: 50 0 j 15 j 15 0 ′ | | ˆ ′ | − | ′ || 20 −10 20 −50 ∆Gjj = 20 log10 Mjj 20 log10 Mref,jj (16) 25 −20 25 −100 ′ |∠ ˆ ′ − ∠ ′ | ∆Pjj = Mjj Mref,jj . (17) −150 30 −30 30 10 20 30 10 20 30 j’ j’ In this example, the average ∆Gjj′ between Mˆ 1 and Mref among all entries is 1.92 dB, and the average ∆Pjj′ among (a) (b) all entries is 9.80 degrees. The average ∆Gjj′ between Mˆ 2 and Mref among all entries is 2.21 dB, and the average ∆Pjj′ among all entries is 10.87 degrees. dB degrees

20 150 In order to furthermore study how the discrepancies differ 5 5 100 10 among modes, the Monte Carlo simulation [24] with suffi- 10 10 50 ciently large number of realizations of the simulation in Fig. 6 0 j 15 j 15 is conducted. Fig. 8 shows the average values of the amplitude 0 20 −10 20 −50 discrepancy and the phase discrepancy on all realizations, −100 ¯ ¯ 25 −20 25 denoted as ∆Gjj′ and ∆Pjj′ , respectively. In this example, the −150 30 −30 30 ˆ ˆ 10 20 30 10 20 30 CM 4 is assumed; M1 and M2 are de-embedded by the ideal j’ j’ spherical array with AC 1 and the virtual spherical DRA array with AC 4, respectively. As can be observed from the figures, (c) (d) while some estimated higher modes tend to have discrepancy with the reference, most of the estimated lower modes are dB degrees reliable. The discrepancies in some higher modes probably 20 150 result from the dominant modes truncation in our proposed 5 5 100 10 approach, as was described in Subsection A. Fig. 9 shows the 10 10 50 ¯ ′ ¯ ′ 0 average values of ∆Gjj and ∆Pjj over all the modes for all j 15 j 15 0 the possible simulation combination, respectively. The results 20 −10 20 −50 show the acceptable average discrepancies, hence prove the 25 −20 25 −100 effectiveness of our proposed approach in general situations. −150 30 −30 30 10 20 30 10 20 30 Since the higher order spherical vector waves are instable for j’ j’ small arrays, AC 3 and AC 6 work relatively better than the rest cases. This is because AC 3 and AC 6 have bigger array (e) (f) sizes than the other cases. 2) Evaluation with Uncertainty: Uncertainties of the de- ˆ ˆ signed spherical array can come from the impact of cables, the Fig. 7. A comparable example of the de-embedded M1, M2 and reference M : (a) and (b) show the amplitude and phase of Mˆ , respectively; (c) and antenna alignment mismatch, the effects of equipment fixtures, ref 1 (d) show the amplitude and phase of Mˆ 2, respectively; (e) and (f) show the and so on. Those factors could affect the de-embedding accu- amplitude and phase of Mref , respectively. racy and make the feasible estimation of the mode-to-mode mapping matrix to a more limited scope. Hence, we introduce j2πPr[0,1] uncertainties to true values of Tsph and Rsph so that their and ∠ρr = e , where Pt [0, 1] and Pr [0, 1] denote the practical realizations are defined as: matrices whose elements are uniformly distributed variables ′ between 0 and 1. The dimensions of P [0, 1] and P [0, 1] ⊙ t r Tsph = Tsph + Tsph ρt (18) are the same as the dimensions of ρt and ρr respectively. ′ Hence both ρ and ρ have random phase factors between 0 R = Rsph + Rsph ⊙ ρr t r sph to 2π. We give different levels of amplitude to each element where the dimensions of ρt and ρr are the same as the of ρt and ρr, and analyze accuracies of the de-embedded Mˆ 2 j2πPt[0,1] dimensions of Tsph and Rsph respectively. ∠ρt = e by using the virtual spherical DRA array. Fig. 10 (a) and IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 8

dB degrees 7 30 6 30 CM 4, AC 4 CM 4, AC 4 5 CM 4, AC 5 CM 4, AC 5 5 5 25 6 CM 3, AC 4 25 CM 3, AC 4 10 4 10 20 5 , dB

jj 20 , degrees j 15 ¯ j 15 3 15 G 4 jj ¯ ∆ P

20 20 ∆ 15 2 10 3 25 1 25 5 2 10 −infinity−60 −50 −40 −30 −20 −10 0 −infinity−60 −50 −40 −30 −20 −10 0 30 0 30 0 |ρ |=|ρ |, dB |ρ |=|ρ |, dB 5 10 15 20 25 30 5 10 15 20 25 30 t r t r j' j'

(a) (b) (a) (b)

Fig. 10. Accuracy of de-embedded Mˆ 2 with different uncertainty levels: dB degrees 6 30 (a) average gain discrepancy and (b) average phase discrepancy over all 5 5 realizations and all modes 5 25

10 4 10 20

j 15 3 j 15 15 and do not rely on the size of the spherical array for de- 20 2 20 10 embedding either, as long as the conﬁguration satisﬁes our 25 1 25 5 proposed conditions.

30 0 30 0 5 10 15 20 25 30 5 10 15 20 25 30 j' j' IV. PROPAGATION REPRODUCIBILITYBY USING DE-EMBEDDED Mˆ (c) (d) The de-embedded Mˆ can be used to reproduce the radio channel transfer functions given target antennas at link ends. Fig. 8. Average values of amplitude discrepancies and phase discrepancies between estimated and referenced mode-to-mode mapping matrix on all In this section, the de-embedded mode-to-mode mapping ma- ¯ realizations of Monte Carlo simulation, under CM 4: (a) and (b) are ∆Gjj′ trix is evaluated in terms of the reproduced channel transfer ¯ ˆ ¯ and ∆Pjj′ , respectively, between M1 and Mref ; (c) and (d) are ∆Gjj′ and function. ¯ ˆ ∆Pjj′ , respectively, between M2 and Mref . A. Simulation Procedure

dB degrees The target antennas considered in Table IV are basically 12 AC1 2.2 AC1 two types: i) the polarized single dipoles which are located at AC2 AC 2 11 2 the origin of the global coordinate, TA 1 - TA 4; ii) the two- AC3 AC 3 10 element dipole array with polarization and spatial diversity, TA 1.8 AC4 AC 4 5 and TA 6, where the elements’ locations are along x-axis 9 AC 5 1.6 AC5 and symmetric about the coordinate origin. First, we compare 8 AC 6 AC6 the reproduced channel transfer functions with the reference. 1.4

CM1 CM2 CM3 CM 4 CM1 CM2 CM3 CM4 Second, we evaluate the performances of different antennas in the same propagation environment. Moreover, we evaluate the performance of propagation reproducibility over the increase of (a) (b) the element spacing in TA 5 and TA 6. Hence we can observe how the correlation changes. Moreover, we can analyze the Fig. 9. Amplitude and phase discrepancies between estimated and referenced target antennas to ﬁnd out, what their greatest acceptable mode-to-mode mapping matrix for all the possible simulation combination: (a) ¯ ¯ electrical size are. The simulation process is shown in Fig. 11. Average value of ∆Gjj′ on all modes; (b) Average value of ∆Pjj′ on all modes. We simulate by Monte Carlo method with a sufﬁciently large number of realizations, namely 1000 in this case.

(b) show the examples where different configurations of the virtual spherical DRA array and different channel models are B. Numerical Example considered. When configuration is AC 4 and channel model is 1) Envelope: Fig. 12 gives an example of the CDFs of CM 4, the proposed de-embedding approach is robust when the the envelopes of the reproduced and the referenced channel element-wise uncertainty level is no more than −20 dB. Here transfer functions. The reproduced channel transfer functions the robustness indicates that both the average gain and average are obtained from Mˆ 1, de-embedded by using AC 3, and Mˆ 2, phase discrepancies over all realizations and all modes start to de-embedded by using AC 6. The target antennas are TA 1 - TA converge. The figure also indicates a discrepancy when either 4, hence the resulting channels are single-input single-output. the AC or the CM is changed. It is observed that, the robustness The Kolmogorov-Smirnov Goodness-of-Fit test [25] with the of the proposed approach does not rely on the channel models, 5% statistics assumption is used to check the similarity of the IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 9

TABLE IV. EXAMPLESOF TARGET ANTENNA (TA)

Transmit Antenna Receive Antenna 0 0 10 10 TA 1 Vertical half-wave electric dipole Vertical half-wave electric dipole |H | 1,VV (V) (V) |H | TA 2 Vertical half-wave electric dipole Horizontal half-wave electric dipole −1 2,VV −1 |H | (V) (H) 10 |H | 10 1,VH ref,VV |H | TA 3 Horizontal half-wave electric dipole Vertical half-wave electric dipole 2,VH (H) (V) |H | −2 −2 ref,VH TA 4 Horizontal half-wave electric dipole Horizontal half-wave electric dipole 10 10 (H) (H) 0 20 40 0 20 40

TA 5 two vertical small electric two vertical small electric 0 0 dipoles separated by ∆d dipoles separated by ∆d 10 10 TA 6 a vertical and a horizontal electric a vertical and a horizontal electric small dipoles separated by ∆d small dipoles separated by ∆d −1 |H | −1 |H | 10 1,HV 10 1,HH |H | |H | 2,HV 2,HH |H | |H | −2 ref,HV −2 ref,HH 10 10 0 20 40 0 20 40

(a)

dB dB VV 20 VV 20

15 15 VH VH

10 10 HV HV 5 5 HH HH

0 0 CM 1 CM 2 CM 3 CM 4 CM 1 CM 2 CM 3 CM 4

(b) (c)

Fig. 12. (a) CDF of envelopes of the reproduced and the referenced channel transfer functions in CM 4; (b) Outage envelope level of H1 at CDF level of −2 −2 Fig. 11. Simulation process for antenna-channel recombination 10 ; (c) Outage envelope level of H2 at CDF level of 10 .

CDFs. As is shown in Fig. 12 (a), the reproduced channel transfer function with dipoles having different orientations at transfer functions agree perfectly with the references, and the the link ends. agreements were observed for other TAs and CMs as well. In 3) Correlation: When the target antennas are TA 5 or TA 6, Fig. 12 (b) and (c), the outage envelope levels corresponding to the channel transfer function with the dimension 2 × 2 can be the CDF level of 10−2 are plotted. The higher the outage level, reproduced. The correlations of the channel transfer function the better the performance. As can be observed, for CM 1, the are defined as: antenna performances are ranked from high to low as TA 1, TA H Ct = H H, (20) 4, TA 3, TA 2; the reproduced H1 and H2 can both reflect the H performance ranking that is the same as the reference. For CM Cr = HH 4, the ranking from high to low is: TA 1, TA 3, TA 2, TA 4, which is different from CM 1. It is obvious that the antennas’ First, the CDFs of the reproduced and the referenced correla- performance vary according to the propagation environments, tions on the Tx side are can be obtained from Monte Carlo and hence, the best performing antenna. simulations. Fig. 14 shows the correlation at 80% probability 2) Gain and Phase Discrepancies: The mean amplitude against varying antenna separation distance ∆d. It is found and phase error of the reproduced channel transfer functions that when the ∆d changes from 0.5λ to 1.3λ, the correlations comparing with the reference, are defined as: of the reproduced channel transfer function coincide with the reference. When the ∆d becomes larger than 1.3λ, the ′ | | | − | || ∆G = 20 log10 H1 20 log10 Href (19) correlations, especially in CM 4, become very different from ′ ∆P = |∠H − ∠H | the referenced value. This result also shows that the electric 1 ref size of the target antenna should be no larger than 1.3λ when Their mean values over all the realizations are shown in Mˆ is de-embedded from AC 3 or AC 6 where the radius Fig. 13. The results show no significant discrepancies, whence of spherical array is λ; this fact coincides with the synthesis the de-embedded Mˆ can successfully reproduce the channel criteria (7). IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 10

degrees VV−VV VH−VH VV−VV VH−VH dB 0.5 0.5 0.5 0.5 VV 1.4 VV 8 1.2 0.4 0.4 0.4 0.4 VH VH 1 6

0.3 (1,2) 0.3 0.3 0.3 (1,2) t (1,2) (1,2) t t t

0.8 C C H H H C H HV HV 1 1 C 1 1 0.6 4 0.2 H 0.2 H 0.2 H 0.2 H 2 2 2 2 H H H H HH 0.4 HH ref ref ref ref 0.1 0.1 0.1 0.1 λ 0.2 2 0.5 1 1.5 2*λ 0.5 1 1.5 2 *λ 0.5 1 1.5 2*λ 0.5 1 1.5 2* CM 1 CM 2 CM 3 CM 4 CM 1 CM 2 CM 3 CM 4 ∆ d ∆ d ∆ d ∆ d

(a) (b) (a) (b)

degrees VV−VV VH−VH VV−VV VH−VH dB 0.5 0.5 0.5 0.5 H VV 1.4 VV 1 8 H 1.2 0.4 0.4 0.4 0.4 2 VH VH H 1 ref 6 0.3 0.3 0.3 0.3 (1,2) (1,2) (1,2) (1,2) t t t t

0.8 H C H C C C H HV HV 1 1 1 0.6 4 0.2 H 0.2 H 0.2 H 0.2 2 2 2 H H H HH 0.4 HH ref ref ref 0.1 0.1 0.1 0.1 λ λ λ λ 0.2 2 0.5 1 1.5 2* 0.5 1 1.5 2 * 0.5 1 1.5 2* 0.5 1 1.5 2 * CM 1 CM 2 CM 3 CM 4 CM 1 CM 2 CM 3 CM 4 ∆ d ∆ d ∆ d ∆ d

Fig. 13. (a) The amplitude and (b) the phase discrepancies between the Fig. 14. Correlations of channel transfer function at 80% probability level reproduced H1 and the referenced Href ; (c) the amplitude and (d) the phase with the increase of ∆d of TA 5 and TA 6: (a) under CM 1; (b) under CM discrepancies between the reproduced H2 and the referenced Href 2; (c) under CM 3; (d) under CM 4

V. CONCLUSION [11] without the conversion to channel response H. Note that in conventional plane wave domain, in order to calculate the This paper proposed a novel antenna de-embedding ap- channel spatial degree of freedom or capacity, the channel proach which extracts the mode-to-mode mapping matrix M model parameters can not be directly used and should firstly defined in the spherical vector wave domain from the channel be converted to H. transfer function that we usually measure in channel sounding. Both the ideal spherical array using tangential dipoles and the virtual spherical DRA array were introduced for the de- APPENDIX A embedding. While the former is used to devise the array SPHERICAL VECTOR HARMONICS configuration that includes the size and the spacing, the latter The spherical vector harmonics are defined as follows: is a practical implementation. The proposed approach was validated numerically under |m| −jmP¯n (cos θ) various propagation environments with various configurations y (θ, ϕ) = q [ · θˆ (21) 1mn mn sin θ of the spherical array. While the estimated lower modes agree | | dP¯ m (cos θ) with the reference perfectly, the discrepancies mainly occur − n · ϕˆ ] on the higher modes. It was also found that the robustness of dθ the proposed approach against uncertainties of spherical array coefficients does not rely on the channel models or the size of |m| the spherical array. Furthermore, the proposed approach was dP¯n (cos θ) y (θ, ϕ) = q [ · θˆ (22) evaluated in terms of the channel transfer functions obtained by 2mn mn dθ the de-embedded M and the assumed target antennas at link |m| −jmP¯n (cos θ) ends. As a result, it was demonstrated that the performance + · ϕˆ ] of different antenna types under different environment is sin θ reproducible by the estimated M, which is beneficial for the √ antenna optimization in system design. ( )m 2 m −jmϕ The proposed approach is instructive for the practical mea- qmn = − e (23) surement of estimating M. Moreover, the estimated mode-to- n (n + 1) |m| mode mapping matrix can not only reproduce the propagation |m| channel, but also can be directly used to obtain the channel where P¯n (cos θ) is the normalized associated Legendre spatial degree of freedom [9], [10] and the channel capacity function defined in [5, p. 318]. IEEE TRANSACTION ON ANTENNAS AND PROPAGATION, VOL. XX, NO. XX, XXXX 2015 11

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