URSI GASS 2020, , , 29 August - 5 September 2020

Phaseless near-field techniques from a random starting point

Rocco Pierri(1), Giovanni Leone(1), Raffaele Moretta(1) (1) Dipartimento di Ingegneria, Universita’ della “Luigi Vanvitelli”, Aversa (CE), 81031, Italia

Abstract Such current radiates an electric field E(x,z) in an homo- geneous medium with wavenumber β. Denote by E(x,z) In this paper we address the problem of reconstructing the the x component of the electric field, and suppose to far field radiated by focusing sources from the knowledge observe |E(x,z)|2 over P lines located at the distances of square amplitude samples of the radiated field on more z1,z2,...,zP which extend along the x axis over the inter- measurement lines in near zone. We show that it is possible vals [−Xo1,Xo1],[−Xo2,Xo2],...,[−XoP,XoP] respectively. to estimate the angular sector where the far field is signifi- The problem we address in this paper is the reconstruction cantly different from zero. This allows to adopt an efficient of the radiation pattern starting from the knowledge of the procedure for the reconstruction of the far field without the square amplitude samples of the electric field E on the ob- use of prior information about it. Such procedure exploits servation domains described above. the quadratic inversion approach for phase retrieval and it consists of two steps. In the first step, it recovers only those samples of the radiation pattern significantly different from Source Domain Observation Domains zero. Later, it reconstructs all the samples of the radiation X pattern starting from an initial guess which is equal to the 2 X solution of the first step, where the radiation pattern is sig- 1 nificantly different from zero, and zero otherwise. Numeri- a cal results show the feasibility of the technique, and its ef- x 0 -a ficiency in terms of data required for convergence. -X 1 -X 1 Introduction 2

Near field to far field methods are usually exploited to re- 0 z z construct the radiation pattern of an antenna. Such tech- 1 2 niques results very useful since they allow to estimate the z far field starting from a set of measurements that can be per- Figure 1. Geometry of the problem formed in restricted domains like lab environments. How- ever, in many cases it is not possible to perform accurate phase measurements hence it arises the need to develop In order to formulate the problem from the mathematical phaseless near field to far field techniques [1, 2]. A typical point of view, now we derive a suitable field expansion. For way to reconstruct the far field pattern from phaseless data the considered 2D geometry, E(x,z) can be expressed as is to tackle a phase retrieval problem by using the quadratic Z a jβz inversion method [3]. Such technique is based on the mini- E(x,z) = H2βRJ (x0)dx0 (1) 4R 1 m mization of a quartic functional and therefore it may contain −a trap points, like local minima and saddle points with null 2 where H1 is the Hankel function of second kind and first gradient. However, the problem of traps can be overcome order, and R(x0,x,z) = p(x − x0)2 + z2. by enlarging the dimension of data space by an increasing In far zone, the x component of the electric field can be of the number of scanning [4]. In order to reduce the num- simplified as follows ber of scanning surface, in this paper we develop a proce- s dure for the reconstruction of the far field from phaseless  2 − jβr 1 p j π u e near field data that can be used for focusing sources. E∞(r,u) = − β e 4 1 − √ I(u) (2) 2 β 2πr √ 2 Mathematical formulation where r = x2 + z2, u = β sinθ with θ indicating the angle with the x axis, I(u) is the magnetic radiation integral and R a j ux 0 0 0 Let us consider a magnetic current J m(x) = Jm(x)yˆ sup- given by I(u) = −a e Jm(x )dx . ported over the interval [−a;a] of the x axis and directed The Fourier transform relation between the magnetic cur- along the y axis that is the axis of invariance (see fig. 1). rent Jm and the magnetic radiation integral I suggests the Fourier harmonics as a possible set of functions expanding Since the functional is not quadratic, it may suffer of the the magnetic current Jm. Here, we assume that only a finite traps problem. However, as shown in [4], the presence of number of harmonics are sufficient to represent the mag- traps is related to the essential dimension of data. The latter n 0 N0 − j π x 0 can be evaluated by introducing a linear representation of netic current, hence Jm(x ) = ∑ I(un)e a . n=−N0 the data, and by computing the number of significant singu- By substituting the truncated Fourier expansion of Jm into (1), we obtain the following representation of the lar values of the linear operator which represents the data. In order to obtain a linear representation of the data, let us x component of the radiated field rewrite the square amplitude samples of the radiated field on the p th measurement line in the form N0 E(x,z) = I(u )ψ (x,z) (3) ∑ n n 2 +N0 +N0 +N0 n=−N0 M2 = I(u )ψ = I(u )I(u )∗ ψ ψ∗  p ∑ n pn ∑ ∑ n l n l jβz a nπ 0 n=−N0 n=−N0 l=−N0 R 2  − j a x 0 where the functions ψn(x,z) = 4 −a H1 βR e dx (8) are called pseudosampling functions. Note that the latter where is the n + N + 1 th column of the matrix L . ψ pn 0 p can be regarded as the Fourier Transform of the restriction By introducing a mapping from each couple (n,l) into a 2  to the set [−a,a] of H1 βR . For this reason they can be different integer q, it is possible to recast (8) under efficiently computed from the numerical point of view by a 2 FFT algorithm. (2N0+1) ∗ 2 Xq = I(un)I(ul) The introduction of the field expansion (3) allows us to for- M = X Ψ where ∗ (9) p ∑ q q Ψ =  mulate the problem of the reconstruction of the radiation q=1 q ψn ψ l pattern from the square amplitude samples of the radiated 2 field. In fact, it is possible to express the square amplitude The equation (9) can be rewritten as Ap(X ) = Mp where functions of the radiated field on the measurements lines as Ap is the linear operator which maps the unknown vector T 2 X = [X ,X ,...,X 2 ] into the data M collected on the 2 1 2 (2N0+1) p N0 2 p th measurement line. By introducing the linear operator E(x,zp) = I(un)ψn(x,zp) p = 1,...,P (4) T ∑ A (X ) = [A (X ),...,AP(X )] , it is possible to represent n=−N 1 0 the data by the linear representation Hence, by discretizing (4), we obtain a set of P subsystems 2 T 2N0+1 A (X ) = M (10) in the unknown x = [I(u−N0 ),....,I(uN0 )] ∈ C which can be expressed as Then, the essential dimension of the data space can be eval- |L x|2 = M2 |L x|2 = M2 ... |L x|2 = M2 (5) uated by computing the singular values decomposition of 1 1 2 2 P P A , and by counting the singular values of A upon a thresh- where ∀p = 1,...,P old dictated by the noise level. ) |L x|2 = L x L x∗ According to [4], local minima disappear if the essential 1 p p p with indicating the ∗ dimension of data is high enough. Despite this, in same Hadamard product and L x denoting the conjugate of p cases the number of data that ensures the lack of traps can (L x) p ; be quite high. With the aim to reduce the number of data, in 2) L ∈ CMp ×(2N0+1) is a matrix whose m th column is this paper we restrict the attention to the reconstruction of p the far field radiated by the wide class of focusing sources. given by [ψ (x ,z ),ψ (x ,z ),...,ψ (x ,z )]; −N0 m p −N0+1 m p N0 m p In fact, for these sources it is possible to use an efficient 2 Mp 3) Mp ∈ R is the vector defined as procedure for phase retrieval. Such procedure allows to re- 2  2 2 2  the number of data required for convergence since it Mp = |E(x1,zp) ,|E(x2,zp) ,...,|E(xMp ,zp) . exploits some information about the radiation pattern that By defining L = [L ,...,L ]T and M2 = [M2,...,M2 ]T it is can be deduced directly from the phaseless data. 1 P 1 P possible to rewrite the set of P subsystems (5) under 4 The estimation of the field direction |Lx|2 = M2 (6) In order to use the phase retrieval technique to be described The problem of finding x from the phaseless data M2 falls in section V, we need to know the angular sector where into the realm of phase retrieval. the radiation pattern is significantly different from zero or, equivalently, ( since u = β sinθ ) the set [umin,umax] where 3 The quadratic inversion method I(u) is significantly different from zero. Such information can be deduced directly from the phaseless data as follow. A solution of such problem can be found by using the The magnetic radiation integral Im(u) is related to the field quadratic inversion method [3], which consists in the mini- at z = zp by the equation mization of the quartic functional Z +∞ 2 j wzp j ux Φ(x) = |Lx|2 − M2 (7) I(u) = e E(x,zp)e dx (11) −∞ 2 (0) Consequently, the Fourier transform of |E(x,zp)| is (7) starting from an initial guess vector x whose main components are xrel and zero the other ones. In such case, Z +∞ ∗ 2 jux − j wzp  − j wzp  even if the functional may exhibit trap points, the starting |E(x,zp)| e dx = I(u)e ∗ I(−u)e −∞ point is in the attraction region; therefore the minimization (12) scheme reaches the global minimum of the functional. By virtue of (12), we can state that the support of the ampli- 2 tude spectrum of |E(x,zp)| has an extension that is twice 6 Number of data required for convergence than the support extension of I(u)e− j wzp . Hence, by com- puting the Fourier transform of |E(x,z )|2 we can derive p In this section, with reference to a particular focusing the support extension of the function I(u)e− j wzp , and con- source, we will estimate the minimum value of the ratio sequently, we can have also a rough idea of the support ex- between the essential dimension of data (M) and the num- tension of I(u). However, since |E(x,z )|2 is a real valued p ber of real unknown (N ) which allows to reconstruct the function, its spectrum is a Hermitian function. As a con- r actual solution of the problem by the above described pro- sequence, the amplitude spectrum of |E(x,z )|2 is always p cedure. In particular, we want to show how the ratio M/N an even function centered at u = 0. For this reason, the r increases with N . As a test case, we consider the recon- amplitude spectrum of |E(x,z )|2 allows to know only the r p struction of the magnetic radiation integral I(u) (shown in width of the set [u ,u ] where I(u) is different from zero 1 2 fig. 2) corresponding to but it does not permits to know the set [u1,u2]. In other 2   words, the amplitude spectrum of |E(x,zp)| contains only π j umax x Jm(x) = cos x e with (16) the information about the size of angular sector where the 2a radiation pattern is relevant but it does not allow to com- with umax = 2, and x ∈ [−a,a]. pute such an angular sector. Nevertheless, a rough estima- tion of the main beam direction in far zone can be obtained 2 0 by observing where |E(x,zP)| attains its maximum, say for Samples of | I(u/ ) | x = xPmax. Hence, we can assume that the direction of main | I(u/ ) | beam in far zone is approximately given by -10

θmax ≈ arctg(xPmax/zP) (13) ) | -20 Consequently, |I(u)| attains its maximum at umax = β sinθmax. | I(u / -30 5 Description of the procedure

Once the set [umin,umax] on which I(u) is relevant has been -40 estimated, it is possible to use the following efficient pro- cedure for phase retrieval. The procedure consists of two -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 u/ steps. In the first step, since some far field samples have low amplitude, the size of the unknown vector is reduced Figure 2. Radiation integral, and samples of the radiation by neglecting the quasi-zero components of x. Hence, the integral to be reconstructed for a source of extension 20λ. N1 new unknown vector xrel ∈ C contains only the relevant N components of x ∈ C with N = 2N0 +1. Consequently, the The square amplitude samples of radiated field are sam- problem to be solved in the first step can be expressed as pled with a uniform sampling step ∆x = λ/4. The exten- sion of the measurement lines along the x axis is such that |L x |2 = M2 rel rel (14) the number of significant values of A1, and A2 differ for L L only 1 unit from the case in which the measurements lines where rel can be obtained by deleting the columns of corresponding to the zero or quasi zero components of the are unbounded [5]. The behavior of the square amplitude vector x. The solution of the quadratic system can be found functions of the radiated field on the measurement lines for by minimizing the functional a source of extension 2a = 20λ is shown in figure 3. In 2 such case, the maximum point of |E(x,z2)| is located at 2 2 2 (x) = |L x | − M x2max = −1.86λ. Since z2 = 6λ, from the equation (13) it Φrel rel rel (15) follows that θ˜max = −0.301rad, henceu ˜max/β = sinθ˜max = The advantage of minimizing the quartic functional (15) in- −0.296. Such value is very close to the actual value of stead of (7) is that in the first case the number of unknowns umax/β = −0.318rad. is lower (by definition N1 < N), and, consequently, also the Since Nr is related to the dimension of the source by ratio between the essential dimension of data and number of the equation Nr = 2N = 8a/λ, in order to establish how unknowns is more favorable. For this reason, it is easier to the ratio M/Nr that ensures the convergence changes ensure the absence of traps in (15) than in (7). Once that a with respect to Nr we perform a set of simulations for global minimum of the functional (15) has been found, the each source of dimension 2a/λ belonging to the set second step of the procedure is to minimize the functional [5, 10,15, 20,25, 30,35, 40,45, 50,55, 60,70, 80,90, 100]. 0 2.5 II step of the procedure -10 2.25

-20 2 -30

2 1.75 -40 |E|

-50 1.5

-60 0 50 100 150 200 250 300 350 400 2 -70 |E (x)| for z =3

1 Dim. of data / Num. unknowns Number of unknowns |E (x)|2 for z =6 -80 2 2.5 -40 -30 -20 -10 0 10 20 30 40 I step of the procedure x 2.25 II step of the procedure Figure 3. Square amplitude functions of the electric field 2 in dB for z1 = 3λ, and z2 = 6λ. 1.75

In particular for each value of the size source, we perform 1.5 a first set of 50 simulations where we search for a number 0 10 20 30 40 50 60 70 80 90 100 of real unknowns Nr1 < Nr. If the minimization scheme re- Dim. of data / Num. unknowns Extension of the source (2a) constructs the unknown sequence in all the 50 test cases, we assume that the ratio between the essential dimension Figure 4. a) Ratio M/Nr in the second step of procedure. of data and the number of real unknowns is high enough b) Comparison of the ratio between the essential dimension to ensure the absence of the traps in the functional, and so of the data and the number of real unknowns in the first and we pass to the second step of the procedure. Otherwise, we in second step of the procedure. The essential dimension enlarge the ratio dimension of data - unknowns by increas- of data is computed by counting the normalized singular ing the number of observation circles and, we repeat the values of the linear operator A upon the value 10−1.5. set of 50 simulations until we found a configuration whose ratio dimension of data - unknowns guarantees the recov- ery of the unknown vector in all the test cases. In all the knowns to attain the actual solution of the problem. simulations the initial guess is chosen at random according to a complex uniform distribution whose components have References both the real and the imaginary part supported on the set [−500, 500]. From this numerical analysis, we find that: [1] G. Junkin, “Planar Near-Field Phase Retrieval Using 1) in all the considered case 2 measurement lines suffice to GPUs for Accurate THz Far-Field Prediction,” IEEE ensure the convergence; Trans. Antennas Propag., 61, 4, April 2013, pp. 1763- 1776, doi: 10.1109/TAP.2012.2220324. 2) the ratio between the essential dimension of data and the number of real unknowns varies according to the diagram [2] A. Paulus, J. Knapp and T. F. Eibert, “Phaseless in fig. 4a. Near-Field Far-Field Transformation Utilizing Com- As it can be seen from fig. 4a, the ratio M/Nr that allows binations of Probe Signals,” IEEE Trans. Antennas to reach the solution, changes very slow with Nr. In fig. Propag., 65, 10, Oct. 2017, pp. 5492-5502, doi: 4b the value of the ratio between the essential dimension 10.1109/TAP.2017.2735463. of data and the number of unknowns in the first step and in the second step of the procedure is compared. As it can [3] R. Moretta, and R. Pierri, “Performance of phase be noted from the figure, in the first step of the procedure retrieval via Phaselift and Quadratic Inversion in such ratio is higher or equal to that of the second step. This circular scanning case,” IEEE Trans. Antennas behavior makes the problem of finding a global minimum Propag., 67, 12, Dec. 2019, pp. 7528-7537, doi: of (15) easier than finding a global minimum of (7). 10.1109/TAP.2019.2930127. [4] T. Isernia, G. Leone and R. Pierri, “Radiation pattern 7 Conclusion evaluation from near-field intensities on planes,” IEEE Trans. Antennas Propag., 44, 5, May 1996, pp. 701, In this paper we address the problem of reconstructing the doi: 10.1109/8.496257. far field from phaseless near field data. To this end, we [5] M. A. Maisto, R. Solimene, and R. Pierri, “A valid an- introduce a phaseless near field to far field method that al- gle criterion and radiation pattern estimation via sin- lows to reconstruct the radiation pattern of focusing sources gular value decomposition for planar scanning," IET starting from a random initial guess. The main advantage Microw. Antennas Propag., 13, 13, October 2019, pp. of the such technique is that it requires a low ratio between 2342-2348, doi: 10.1049/iet-map.2018.5240. the essential dimension of the data and the number of un-