Abstract â Spherical wave modeling has gained interest in radio propagation channel modelling. The idea is to consider the responses of radio channels for ...
Noise Sensitivity Analysis of Spherical Wave Modelling of Radio Channels using Linear Scanners Afroza Khatun, Tommi Laitinen, and Pertti Vainikainen Aalto University School of Science and Technology, Department of Radio Science and Engineering, SMARAD, P.O. Box 13000, 00076 AALTO, Finland {afroza.khatun,tommi.laitinen,pertti.vainikainen}@tkk.fi Abstract — Spherical wave modeling has gained interest in radio propagation channel modelling. The idea is to consider the responses of radio channels for different spherical wave mode fields. In this paper, generation of spherical wave mode responses of a channel from measured data over a cubical scan surface is studied. Based on virtual array principle the robustness of spherical wave generation over 2-octave band for different cubical scan schemes is analyzed. This method presents the advantages for exploiting the measured data from widely used 3D linear scanners. Importantly, linear scanning supports well the acquisition of cubical scan data. Noise sensitivity analysis on a wide frequency range shows the feasibility of cubical scanning. Rules for the required number of scan locations on a cubical surface are provided. Index Terms — Antenna arrays, scanning antennas, noise, sensitivity, spherical wave expansion.
I. INTRODUCTION Spherical vector wave mode expansion (SWE) is a set of canonical solutions of the Maxwell’s and Helmholtz equations in spherical coordinates. It has been well understood [1] and widely used in spherical near-field as well as far-field antenna measurement. Recently, research has been performed on spherical wave modelling on radio wave propagation channels [2] instead of the usual plane-wave approach which has been used comprehensively in the field of array signal processing, channel parameter estimation and channel modelling. Basically, the idea in spherical wave modelling is to consider the responses of the radio channel for different spherical wave mode fields. In mathematical terms if the channel is described as the 𝑀 matrix [2] (assuming a single frequency case), then the relation between spherical mode coefficients of the radiated field of the transmit antenna 𝑄 and those of inward propagating spherical fields at the receiving antenna 𝑎 can be written as 𝑎 = 𝑀𝑄. The mode coefficients 𝑎 may be mapped into the received signal at the receiver 𝑤 = 𝑅𝑇 𝑎 through the antenna receiving coefficients 𝑅 [1]. Thus, the received signal can be determined with the known radiation patterns of receiver and transmitter antenna, and the known channel matrix 𝑀. In this paper the focus is the development of a proper measurement method that allows the determination of the channel matrix 𝑀 in a robust way. Using virtual array principle, a common way of radio channel modelling is based on radio channel measurement with, as an illustration, spherical scanning and it would be evidently the optimum
way to perform the scanning for the purposes of generating spherical wave modes. Although in the case of spherical scanning the spherical wave generation may be robust and stable, spherical scanning for the channel measurement purposes is impractical in some situations compared to, for example, the linear scanning. Apparently, linear scanners are very widely used in laboratories, and they clearly provide a possibility for both practical and time-efficient scanning of large volumes. In our previous paper [3], we focused on spherical wave generation in such a case, where substitution of the AUT (antenna under test) is done with a set of Hertzian dipole sources distributed over a cubical scan surface enclosing the antenna. We applied the idea of creating a virtual array with dipole sources to find out such a set of excitations of the dipole sources that this virtual array radiates any desired single spherical mode. We showed that generation of all desired spherical wave mode fields is possible accurately by using a single probe, which is electric Hertzian dipole probe in our work, by performing the cubical scanning for three orthogonal probe orientations (x, y, z). The details of the cubical scanning and data extraction based on spherical wave mode expansion are reported in [3]. In our previous work, we did not consider the noise and measurement error, and the influences of the frequencies on the cubical scanning using Hertzian dipole. In the present paper, the analysis presented in [3] is complemented. The cubical scanning is studied on a wide, 2octave frequency range. The noise sensitivity is analyzed by introducing certain noise level in the excitation voltages of the dipole probes of the virtual array on the cubical scan surface. The purpose of this paper is to investigate the noise sensitivity of the cubical scanning over a frequency range and provide also rules what the number of measurement locations for cubical scanning must be to reach a desired level of accuracy for the generation of spherical wave mode fields. The cubical scanning principle could be applied for examining the influence of the radiation pattern of a sensor application on ToA (time of arrival) estimation in an environment of Ultra-wideband (UWB) systems, like what has been done, for example, in [4]. II. SPHERICAL WAVE EXPANSION In a source free region and outside the minimum sphere of radius 𝑟0 enclosing the source (AUT), the spherical wave
expansion of the propagating electric field 𝐸 𝑟, 𝜃, 𝜑 in spherical coordinates (𝑟, 𝜃, 𝜑) system is expressed in terms of a truncated series of spherical vector wave functions [1] as 2
𝑁
𝑛 3
𝐸 𝑟, 𝜃, 𝜑 = 𝑘 η0
𝑄𝑠𝑚𝑛 𝐹𝑠𝑚𝑛 𝑟, 𝜃, 𝜑
1
𝑠=1 𝑛=1 𝑚 =−𝑛
With 𝑁 = 𝑘𝑟0 + 𝑛1
(2)
where 𝑘 is the wave number, 𝜂0 = 𝜇/𝜀 is the free-space field impedance, 𝑄𝑠𝑚𝑛 are the spherical wave coefficients 3 (transmission coefficients) and 𝐹𝑠𝑚𝑛 𝑟, 𝜃, 𝜑 represent the power normalized spherical vector wave functions of the outward propagating fields. The number of radial spherical wave modes 𝑁 depends mainly on the antenna dimensions and the operating frequency [1]. In Eq. (1) 𝑚 = −𝑛 … 𝑛 indicates that a full set in 𝑚 is used and the index 𝑠 denotes either the TE or TM field. In Eq. (2) for 𝑁, 𝑛1 should be chosen to provide sufficient accuracy of the field characterization. For most practical purposes, 𝑛1 = 10 is considered sufficient [1]. For electrically relatively small antennas, for instance, antennas with 𝑟0 < 2𝜆, a clearly smaller value of 𝑛1 could apply, for example, 𝑛1 = 2 [5]. The total number of spherical wave modes as a function of radial modes 𝑁 is 𝐽 = 2𝑁 𝑁 + 2 .
We have shown that the greater numbers of sampling locations are required for larger cube to reach a desired level of accuracy to generate spherical wave mode fields up to a certain spherical wave index 𝑛 = 𝑁. In this paper, we consider similar topology for the cubical geometry with the exception of having increased sampling locations on each side of the cube with 𝑁1 = 5 … 10 . Furthermore, the analysis is made for 𝑎 𝜆 = 0.6 to 𝑎 𝜆 = 2.4, where 𝑎 is the length of the side of the cube. Thus, we consider different cube sizes in wavelengths () over 2octave band. III. SYNTHESIS OF THE RADIATED FIELD A. Excitations of Virtual Array In our work we considered the radiated electric far field of any desired spherical wave mode and only this mode as the target field, denoted as vector 𝐸𝑇 . We then found out such a set of excitation coefficients, denoted as vector 𝐶 for the dipole moments that provide the desired target field. Our approach to find out the excitation coefficients was the matching of the dipole fields to the target field. The details on how to calculate the target field using SWE, how to make the matching and calculation of the complex excitation coefficients for the dipoles are reported in [3]. In a matrix form, the relation between 𝐸𝑇 and 𝐶 is as follows [5]
III. CUBICAL SCANNING SCENARIO In our previous work [3] we investigated the cubical scanning for different sizes of virtual cubes (1 𝜆 × 1 𝜆 × 1 𝜆 and 2 𝜆 × 2 𝜆 × 2 𝜆 ) centered in the Cartesian coordinate system (x, y, z) in a way that each side of the cube was divided into 𝑁1 × 𝑁1 smaller squares of equal sizes and the sampling locations lied on the midpoints of these squares. The cubical scanning was performed with 𝑁1 = 2 … 7 for three orthogonal probe orientations in each sampling location. An example of with 𝑁1 = 5 is shown in Fig. 1.
𝐸𝐶 𝐶 = 𝐸𝑇
(3)
where 𝐸𝐶 is the coupling matrix consists of complex electric fields of all dipoles of the virtual array in two tangential polarizations in the far field [3]. B. Modelling of Noise By solving the above Eq. (3) we find the excitation coefficient vector 𝐶 for the virtual array. In [3] we applied these excitation coefficients on the probes over the cubical surface and generated the synthesized target field vectors 𝐸𝑆 for all spherical wave modes. In practical measurements or scanning scenarios, the measured data is always contaminated by noise, and therefore, reconstructing of any antenna pattern using the virtual array principle in presence of input noise is paramount to its practical success. Hence, starting with the calculated exact 𝐶 , error is introduced in 𝐶 to model noise in the measurement. A random noise is introduced on vector 𝐶 element by element so that the error due to noise varies from element to element as follows 𝐶𝑁 = 𝐶 + 𝜌 × 𝑒 𝑗 2𝜋𝑃 [0,1] × 𝐶 .
(4)
Here vector 𝐶𝑁 consists of the noisy excitation coefficients, 𝜌 is the noise amplitude, 𝑒 𝑗 2𝜋𝑃 [0,1] is random phase factor between 0 and 2𝜋 . Then we synthesize the target field, Fig. 1 Cube geometry for 𝑁1 = 5
𝐸𝑁,𝑆 = 𝐸𝐶 𝐶𝑁
(5)
We repeat the above calculation process for all the modes for 𝑗 = 1 … 𝐽 for all cases of 𝑁1 = 5 … 10 for 𝑎 𝜆 = 0.6 … 2.4. IV. RESULTS Due to cut-off properties of the spherical wave series for the outward propagating modes the cube size in wavelengths sets the limit, via Eq. (2), for the highest spherical mode that we can possibly generate. We consider here the cube sizes from 0.6 to 2.4 and as a function of cube size we consider the generation of spherical wave modes up to the truncation number 𝑁 = 4 … 17, that is in line with Eq. (2), where, as 𝑎 mentioned before, 𝑛1 = 2 [5]. Here 𝑟0 = √3 is applied. 2 In this section, the sensitivity analysis of generating the spherical mode fields using cubical scanning is described. We investigate the maximum normalized 𝑛-mode error, 𝜀(𝑛) [3] in the presence of noise, which reflects the maximum error in generating spherical wave modes as a function of 𝑛, and it likewise leads to an insight on how many sampling locations per cube side, 𝑁1 × 𝑁1 , are required to generate any spherical wave mode up to 𝑛 = 𝑁 within the maximum allowable error level. 𝜀 𝑛 = 𝑚𝑎𝑥 𝛿 2(𝑛 − 1 𝑛 + 1 + 1 … 2𝑛(𝑛 + 2)) (6) Here 𝛿(𝑗) = 𝑚𝑎𝑥
𝐸 𝑁 ,𝑆 −𝐸𝑇 max 𝐸 𝑇
is the maximum error between
the target field vector 𝐸𝑇 and the synthesized field vector 𝐸𝑁,𝑆 as a function of spherical wave mode 𝑗 [3]. We calculate 𝜀 𝑛 for 𝑁1 = 5 … 10 for the cube sizes from 𝑎 = 0.6 to 2.4 . In this paper we add −40 dB relative noise level with random phase to the dipole excitations vector 𝐶 , through Eq. (4) to model the noise, that is, 𝜌 = −40 dB and synthesize the target field, 𝐸𝑁,𝑆 using Eq. (5). In Fig. 2 the maximum normalized 𝑛 -mode errors is shown for the case with 𝑁1 = 10 for 𝑎 𝜆 = 0.6 … 2.4.
n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9 n=10 n=11 n=12 n=13
-10
-20
(n)
0
-10 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9
-20
-30
-40
-50
0
-30
-40
-50
-60 0.6 0.72 0.84 0.96 1.08 1.2 1.32 1.44 1.56 1.68 1.8 1.92 2.04 2.16 2.28 2.4
a/
Fig. 2.
The result is quite anticipated. For low values of 𝑛 , for example, for 𝑛 = 1 … 3 the errors 𝜀 𝑛 are generally of the order of −40 dB and for some values of 𝑎 𝜆 even clearly lower than −40 dB. In the range from 𝑎 𝜆 = 0.6 … 2.4, for 𝑛 > 3, the error level depends on 𝑛 and 𝑎 𝜆, so that for a given 𝑛 , with decreasing 𝑎 𝜆 the error 𝜀 𝑛 starts to significantly increase at some point. For example, for 𝑛 = 5 the error 𝜀 𝑛 exceeds 30 dB for approximately 𝑎 𝜆 ≤ 0.9 and for 𝑛 = 7 it does that for approximately 𝑎 𝜆 ≤ 1.5 . The errors are greater than 30 dB for all modes for 𝑛 > 10 (the curves for 𝑛 > 13 are not seen in the figure). It can be observed from the above figure, that actually the (only) significant factor which influences the uncertainty in generating the spherical wave modes up to certain spherical mode index 𝑛 = 𝑁 , is the cube size in wavelengths. The natural explanation for this is the cut-off property of the spherical wave series. When we increase the cube size in wavelengths this cut-off occurs at a higher truncation number 𝑁. But for all the cases where we are clearly below the cut-off, for example, for 𝑁 = 1 … 3, the error level is always of the order of −35 dB in the maximum. In brief, it means that for any arbitrary AUT, which radiation pattern can be characterized by the spherical wave modes truncated by 𝑁 = 3 , may be accurately synthesized by a cubical scanning with the cube size 𝑎 = 0.6 𝜆 … 2.4 𝜆 using an electric dipole probe in three polarizations with 10 × 10 sampling locations on each side of the cube. Now we reduce the number of sample locations form 𝑁1 = 10 to 𝑁1 = 5 and investigate the results. Fig. 3 presents the maximum normalized 𝑛 -mode errors for different cube sizes, 𝑎/𝜆, with 𝑁1 = 5.
(n)
𝐸𝑁,𝑆 for each spherical wave mode separately using the following equation
The maximum normalized 𝑛-mode error 𝑁1 = 10
N=4
-60 0.6 0.72 0.84 0.96 1.08 1.2 1.32 1.44 1.56 1.68 1.8 1.92 2.04 2.16 2.28 2.4
a/
Fig. 3.
The maximum normalized 𝑛-mode error 𝑁1 = 5
We find that the errors are now higher than 0 dB for 𝑛 > 9. By comparing with the case of 𝑁1 =10, Fig. 3 shows worse performance and we can not generally reach −40 dB error level any more in generating spherical wave modes. Hence, the performance of the scanning system, and the requirement for the accuracy determine whether a sampling scheme with 𝑁1 = 5 can be accepted. But if the desired error level is the order of −20 dB in the maximum then we can possibly
generate spherical wave modes truncated by 𝑛 = 𝑁 = 4 for a range with the cube sizes from approximately 𝑎 = 0.72 𝜆 to to 1.44 𝜆. This range is indicated by an arrow in Fig. 3. It is also noticeable that scanning over a cube with a larger than 1.44 with 𝑁1 = 5, we are not, in general, able to generate spherical modes with error less than −20 dB, which indicates that a greater number of sample locations would be evidently required. 0
-10 n=1 n=2 n=3 n=4 n=5 n=6 n=7 n=8 n=9
(n)
-20
-30
-40
-50 N=4 -60 0.6 0.72 0.84 0.96 1.08 1.2 1.32 1.44 1.56 1.68 1.8 1.92 2.04 2.16 2.28 2.4
a/
Fig. 4.
The maximum normalized 𝑛-mode error 𝑁1 = 6
In Fig. 4 we present the maximum normalized 𝑛 -mode error with 𝑁1 = 6. To summarize the results we can say that scanning with 𝑁1 = 6 gives a better performance as compared to the case with 𝑁1 = 5 in the case where modes up to 𝑁 = 4 are desired to be generated with the acceptable error level below −20 dB. It is shown from Fig. 4 that we might now have the possibility to generate any spherical wave mode from approximately 𝑎 = 0.6 𝜆 to 1.68 𝜆 . This range is indicated by an arrow in Fig. 4. It is interesting that for 𝑁1 = 5 or 6 , the results do not follow the similar logical behavior as for 𝑁1 = 10 in which case we have a more predictable output. Hence, a more detailed analysis of different sampling schemes would prove useful. V. CONCLUSION Noise sensitivity analysis for the cubical scanning has been presented. The sensitivity has been studied for the cube sizes 𝑎 𝜆 = 0.6 to 2.4. We have shown that cubical scanning for three orthogonal probe polarizations provides a possibility to generate spherical wave mode fields with an acceptable uncertainly if the scanning scheme is chosen appropriately. The results of this paper provide the rules for choosing the number of measurement locations for cubical scanning to generate spherical wave modes up to a certain spherical mode index 𝑛 = 𝑁. It is shown that the rules for choosing the scan locations are not so straight-forward which gives the motivation for a more detailed investigation of the methodology. The SWE is the precise expression to describe the field behaviour of an antenna. Cubical scanning with
three orthogonal probe orientations can thus be found useful in applications where the spherical mode responses of a channel are required. Future investigation can be extended to deal with the influences of the real probes considering the practical measurement environment. ACKNOWLEDGMENT The work is a part of joint research project between the Tokyo Institute of Technology and Aalto University School of Science and Technology in a Finnish-Japanese research programme funded by the Japan Society for the Promotion of Science and the Academy of Finland. T. Laitinen would like to thank Academy of Finland (decision notification no. 129055) for the financial support of this work. REFERENCES [1] J. E. Hansen, Spherical near-Field Antenna Measurements. UK, London: Peter Peregrinus Ltd., 1988. [2] A. Alayón-Glazunov, “On the Antenna-Channel Interactions: A Spherical Vector Wave Expansion Approach,” Doctoral Thesis, Lund University, Lund, Sweden, February, 2009. [3] A. Khatun, T. Laitinen and P. Vainikainen, “Spherical Wave Modelling of Radio Channels using Linear Scanners,” in Proc. 3rd European Conference on Antenna and Propagation, (EuCAP 2010). 12-16 April 2010. [4] M. Dashti, T. Laitinen, M. Ghoraishi, K. Haneda, J. Toivanen, J. Takada and P. Vainikainen, “Antenna Radiation Pattern Influence on UWB Ranging Accuracy,” in Proc. 3rd European Conference on Antenna and Propagation, (EuCAP 2010). 1216 April 2010. [5] T. Laitinen, “Advanced Spherical Antenna Measurements,” Thesis for the degree of the Doctor of Science in Technology, TKK, Espoo, December, 2005.
