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The Radio Propagation Channel
Using a vector network analyzer (VNA) we can measure the forward voltage gain scattering parameter (S21) from all directions in the room of interest. We can do this by having an antenna rotating mechanically around an axial point. For example, by measuring the S21-parameter on the surface of a sphere. If the antenna radiation pattern is known the room’s spatial properties can be extracted from the measurement. This approach will give us a direct estimation of Angle-of-Arrival (AoA). Hence, when we point the antenna in the direction of an incoming wave the estimation from that direction directly gives us the scattered wave.
There are many ways to approach the problem. For example measuring the S-parameters on some other surface or volume than a sphere. We have also decided to limit the test to estimate the AoA and complex amplitude property. The main motivation for this is time and also we feel that this will provide enough data to discuss within the scope of this master thesis.
Propagation Channel Model
When an electromagnetic signal is sent out from a transmitting antenna it enters the propagation channel. The signal during it’s path undergoes different propagation mechanisms, such as scattering, reflection and diffraction. Hence, the received signal will arrive from different paths. By superposition, the received signal will be a sum of all versions of the transmitted signal impinging from different directions. The receiving angle is named the AoA. The angle of the transmitted signal is called Angle-of-Departure, (AoD). Assuming that the waves are scattered from objects in the far-field region of the receiving antenna, the waves can be approximated as plane waves. The incident field thus can be written as a sum of plane waves incoming from different directions Nwaves Einc(θ, φ) = X Ene−jkn•rn+αn , (2.1) where φ is the spherical azimuth angle, θ the spherical elevation angle, En is the complex amplitude and kn • rn builds up a constant phase front that propagates in the direction of kn.
Since the waves have taken different paths they will imping at the receiver with different amplitudes, phases, polarizations and time delays. This leads to a constructive and destructive interference called multi path fading.
Figure 2.1: Illustration of a multi path. TX is the transmitting antenna and RX is the receiving antenna.
Far-Field Region
In the far-field region of an antenna, also called the Fraunhofer region, the electro-magnetic field is built up by plane waves. The radiation pattern does not change it’s shape as a function of the distance. There are three criteria[7, p.167] that are fulfilled in the far-field:
R>>D (2.2)
R>>λ (2.3)
R > (2D2)/λ, (2.4)
where D is the largest dimension of the antennas and λ is the wave length.
Basic Antenna Properties
Radiation Pattern
The antenna radiation pattern is described as the variation of the radiation intensity of an antenna in the far-field region. The radiation pattern can be presented in terms of either the power or the field pattern[7, p.28]. Throughout this thesis the radiation pattern used is the complex field pattern in far-field, F(θ, φ) [v/m]. The radiation pattern can be written as the angular components of the far-field region electrical field intensity of the antenna as F(θ, φ) = Fφ(θ, φ) + Fθ(θ, φ), (2.5) where Fφ(θ, φ) is the azimuthal components and Fθ(θ, φ) the elevation components.
Gain
The gain of an antenna is defined as ”the ratio of the intensity, in a given direction, to the intensity that would be obtained if the power accepted by the antenna were radiated isotropically. The radiation intensity corresponding to the isotropically radiated power is equal to the power accepted (input) by the antenna divided by 4π.”[23] Which can be expressed as G(θ, φ) = 4π | F(θ, φ) |2 , (2.6) ηPin where η is the intrinsic impedance of the medium and Pin is total input power. We can also define the bore sight gain as G0 = G(θ, φ) |max (2.7)
Half-Power Beam Width
The half-power beam width is defined as ”In a plane containing the direction of the maximum of a beam, the angle between the two directions in which the radiation intensity is one-half value of the maximum of the beam.”[23]
Free Space Path Loss
In free-space, where no obstacles disturbs the signal, between a receiving- and a transmitting antenna the signal power at the receiving antenna is given by Friis transmission formula[7, p.95].
λ 2 Pr = Pt G0tG0r, (2.8) 4πR
where Pr is the received signal power, Pt the transmitted signal power, λ the wave-length, R the distance between the antennas and G0t, G0r the gain of the transmit-ting respectively the receiving antenna. In decibel scale, dB, this formula becomes: Pr,dB = Pt,dB + G0t,dB + G0r,dB + 20log10 λ , (2.9)
Circular Array
As mentioned earlier we will be using a mechanically rotating antenna as a receiver to measure the radio propagation channel. In a static environment this can be viewed as a circular array if the antenna is kept on a fixed radius, see figure 2.2.
If the rotational spacing is equal between the antenna placements we can view the rotational antenna as N identical elements with equal angular spacing. If they are all placed on the radius a and the distance from the rotational center to the observation point noted r the phase difference can be written in the same way as with a circular array [7, p.365-356].
We are now faced with a linear inverse problem. Inverse problems arise when one tries to extract hidden information contained in measurement data. Thus, it is a very important and A¡˜ common mathematical problem. Here, we are dealing with a specific type of inverse problem, namely a deconvolution problem.
There are several potential difficulties involved with solving a deconvolution problem
• Noise in the measurement data v or inaccuracies in the matrix B.
• The matrix B is ill-conditioned.
• The problem is ill-posed.
In our channel model the noise vector, e, is modeled as an additive white Gaus-sian noise(AWGN), e. Which leads to that (2.18) becomes v = Bp + e, (2.21)
The condition number of B gives an indication of how sensitive the solution is to perturbations in B or in the measurement data v. A problem with a large condition number is called ill-conditioned, [19].
cond(B) =k B k2k B−1 k2, (2.22)
When the condition number is large, even if the noise level is low, the noise will have a huge impact on the resulting estimation pˆ. Using (2.22) we can calculate the condition number for the matrix B. For example our model B of the size 720 × 720 the condition value then is 1.1097 × 106. Therefore, we can say that we have an ill-conditioned problem.
Many inverse problems are ill-posed. A problem is said to be ill-posed if it is not well-posed. A well-posed problem has to fulfill the following requirements
• A solution must exist.
• The solution has to be unique.
• A small change in the initial condition should lead to a small change in the solution.
A direction solution of (2.18) cannot rely on the direct inversion since the actual problem is given by (2.21). Hence, the standard approach is to find the solution pˆ that satisfies the following conditions
min k p k2 subject to min k B • p − v k2, (2.23) where k p k2 is the euclidean norm of the solution and k B • p − v k2 is the residual norm.
In the next section we explain how to solve (2.21) by the use of singular value decomposition.
Singular Value Decomposition (SVD)
The singular value decomposition is a factorization form of a matrix into two unitary matrices and one diagonal matrix of singular values.
Let B be the matrix to decompose, then we can write B = UΣV†, (2.24) where † denotes the Hermitian-conjugate. The left and right matrices, U ∈ Cmxm and V ∈ Cnxn , satisfies the unitary condition U†U = I, V†V = I and Σ ∈ Rmxn is a diagonal matrix. The diagonal matrix contains the elements {σ}ni=1 which are given in a descending order. These elements are called the singular values of the matrix B.
The inverse of B can be obtained by inverting the singular values and writing the SVD factors in reverse order [10]. Note that if any of the singular values σi is zero the inverse will be undefined.
B−1 = VΣ−1U†
We refer to this inverse as the naive inverse of B. (2.25) may also be written as sums of n rank-1 matrices [10] where νi is the column vectors of V and µi is the column vectors of U.
Before we continue, let’s study the singular values of B. As mentioned pre-viously, the singular values are given in a descending order. Depending on the structure of matrix B many ill-conditioned problems can be categorized into two groups: rank-deficient- and ill-posed- problems [19].
In rank-deficient problems there exist a gap between the size of the larger singu-lar values and the smaller. The smaller singular values reflects the approximation errors in the model of B. In these cases the problem can be transformed to a well conditioned problem by identifying the matrix’s numerical rank. The rank corre-sponds to the singular value before the gap. The other major group is the case where there is no clear gap. In this case it’s not as easy to identify the rank of the matrix. This case is called ill-posed. Here we instead have to find a middle ground between the residual norm and the size of the solution, see (2.23). Let us plot the singular values from the matrix B. It can be seen in figure 2.4 that identifying a clear gap is not possible. Instead there is a gradual decay. We therefore can say that we have a problem that is ill-conditioned and ill-posed.
The elements of µ†iv are called the Fourier coefficients. A test to see if the so-lution is meaningful is to look at the decay rate of the Fourier coefficients. The Discrete Picard Condition [16] says that the solution is meaningful if the Fourier Coefficients decrease (on average) at a higher rate then the singular values. This also means that the quotient vector µ†iv must not increase. This is rarely the case σi in real measurements because of measurement errors in v and round off errors in the matrix B. Here the Fourier coefficients will settle on a level due to the errors in v and the singular values will settle on a level due to the errors in the matrix B, as seen in figure 2.5a. However, if the underlying exact problem satisfies the Discrete Picard Condition it is often possible to find a meaningful approximation to the solution. In ill-conditioned problems there is a need for some kind of regu-larization. The objective is to modify the system so that the solution becomes less sensitive to the influence of measurement noise and/or inaccuracies in the matrix B. If the regularization of the problem is successful the system for regularized solution satisfies the Discrete Picard Condition. An example of this can be viewed in figure 2.5b.
Truncated Singular Value Decomposition
There are many different methods and variants to regularize the problem (2.23). Two methods of wide-spread use are the Tikhonov regularization and the Truncated Singular Values Decomposition (TSVD). The advantage of TSVD is that it is very intuitive because you just remove the singular values that amplifies the noise in the solution. The solution for different regularization parameters can easily be calculated. The downside to TSVD is that we first have to compute the SVD. This is for large problems very computationally heavy. In those cases other methods can perhaps be a better choice, such as the Tikhonov regularization. In our case the problem is not large. The size of the problem is limited by the measurement time.
Since the purpose of the thesis is not to compare different regularization methods we have chosen to only use the TSVD. The choice is made mainly because of it’s simplicity.
In the Truncated Singular Value Decomposition (TSVD) the problem is filtered by setting the smaller singular values of B equal to zero. Truncating the SVD will lower the condition value of Bk and the solution will be less contaminated by noise. The condition number of Bk is given by κ(k) = σ1/σk.
Let us divide the singular values into two parts: one containing the values that will lead us to the solution of the problem (values 1 to k) and one part containing the values that will distort the solution (the last n−k values). The resulting matrix, called Bk, is defined as
Bk = UΣkV†, Σk = diag(σ1, . . . , σk, 0, . . . , 0) ∈ Rmxn (2.28)
Choosing a Regularization Parameter
Truncating the SVD will directly improve the condition number of B due to the definition of SVD but, on the other hand, too much truncation will remove wanted information from the solution. There has to be a trade off between these two. There are different ways to approach this problem. The most simple way is perhaps to filter the solution so that the residual norm (2.23) equals an upper bound set by the norm of the error in the measurement data, k e k2, where e is the additive noise from v. This is called the discrepancy principle [20]. A problem with this is that we don’t know that error, at least not with enough accuracy. It also tends to filter to much from the solution [18]. A common method to graphically determine this trade off is called the L-curve method. This method extracts information from the noise contaminated measurement data. This is the method we have chosen to use.
Table of contents :
1 Introduction
1.1 Problem statement
1.2 Thesis Outline
2 The Radio Propagation Channel
2.1 Propagation Channel Model
2.2 Far-Field Region
2.3 Basic Antenna Properties
2.3.1 Radiation Pattern
2.3.2 Gain
2.3.3 Half-Power Beam Width
2.4 Free Space Path Loss
2.5 Circular Array
2.6 The Measured Voltage and the Normalized Incident Field
2.7 A Discrete Inverse Problem
2.8 Singular Value Decomposition (SVD)
2.9 Truncated Singular Value Decomposition
2.10 Choosing a Regularization Parameter
2.11 Dealing with the Solution to the Regularized Problem
2.12 Successive Interference Cancellation
2.13 Implementation in MATLAB
3 Simulation
3.1 Simulation Results
3.1.1 One Scatterer Case
3.1.2 Two Scatterer Case
3.1.3 Five and Ten Scatterer Cases
3.1.4 Simulation Comments
4 The Measurement System
4.1 Introduction to The Measurement System
4.2 MOLLY – A St¨aubli RX90L Robot
4.2.1 MOLLY definitions
4.2.2 Work envelope of MOLLY
4.2.3 Precision of MOLLY
4.3 Taxen – A 1-Dimensional Stepper
4.3.1 Precision of Taxen
4.4 Work envelope conclusion
4.5 Vector Network Analyzer
4.6 Measurement Times
4.7 Antennas
4.7.1 The Horn Antenna
4.7.2 The Dipole Antenna
5 Phase Center
5.1 Phase Center Setup
5.2 Phase Center Result
5.3 Phase Center Conclusion
6 The Measurement
6.1 Experiment Setups
6.1.1 Possible error sources
6.1.2 Setup 1
6.1.3 Setup 2
6.2 Measurement and Estimation Results
6.2.1 Setup 1
6.2.2 Setup 2
6.3 Discussion
6.3.1 Setup 1
6.3.2 Setup 2
7 Final Conclusions and Future work
7.1 Final Conclusions
7.2 Future Work
Appendices
