Wigner distribution function approach to analyze MIMO communication within a waveguide

Abstract

Multiple-input-multiple-output (MIMO) communication is a technology to create high capacity wireless links. The main aim of this paper is to provide a foundation to mathematically model wireless chip to chip communication within complex enclosures. This paper mainly concentrates on modelling wave propagation between transmit and receive antennas through a phase space approach which exploits the relationship between the field-field correlation function (CF) and the Wigner distribution function (WDF). A reliable model of wireless chip-to-chip (C2C) communication helps mitigate the information bottleneck caused due to the wired connections between chips, thus, help improve the efficiency of electronic devices of the future. Placing complex sources such as printed circuit board (PCB) inside a cavity or enclosure results in multi-path interference and hence makes the prediction of signal propagation more difficult. Thus, the CFs can be propagated based on a ray transport approach that predicts the average radiated density, but not the significant fluctuations that occur about it. Hence, the WDF approach can be extended to problems in finite cavities that incorporates reflections as well. Phase space propagators based on classical multi-reflection ray dynamics can be obtained by considering the high-frequency asymptotics.

1. Introduction

The main aim of this paper is to analyse MIMO communication within a waveguide and the dynamics of wave propagation when equipped with transmitters and receivers on either ends. It is well known that communication channels are extremely sensitive to frequency changes and other system parameters. The channels in a wireless C2C communication can be assumed to be static, that is, they do not vary over time. In such a scenario, the modelling and analysis of wireless C2C communication under the influence of statistical fluctuations becomes extremely critical. These fluctuations are generated as a result of the system noise caused by the electronic components and interference noise caused by the multiple reflections in an enclosure. The presence of such components, devices and irregular enclosures cause blockage, losses and multiple scattering of the electromagnetic (EM) waves. This creates a complex wireless communication channel between the transmitters and receivers in a MIMO setup. In such complex and multi-reflective environments, the simulation of electromagnetic waves using standard methods such as finite difference time domain (FDTD) and transmission line matrix (TLM) methods [1] are unpracticed as the meshes required to model features on a wavelength scale are too large. Thus, approximate or statistical methods can provide a feasible means to simulate the signals in such scenarios. Approximations such as dynamical energy analysis (DEA) [2], which is similar to the concept of ray tracing, are computationally much easier than the traditional techniques. The ray tracing approximation in free space can be extended to analyse MIMO communication within enclosures. However, here we address the problem by employing an exact method that helps us obtain the qualitative and the quantitative behaviour of EM fields within a waveguide. In our future work, we are interested in obtaining an approximation using concepts in DEA to predict connection strength, the maximum channel capacity in any given complex environment such as reverberant chambers etc. and the number of feasible channels to transfer information through the calculation of number of non-zero eigenvalues along with the incorporation of multi-path propagation.

Furthermore, in the context of MIMO communication, it is important to know the maximum information that can be transferred between transmitters and receivers by restricting the number of antennas in the respective domains within an enclosure. In general we are interested in arbitrarily shaped enclosures but use the simplified geometry of a waveguide in this paper to explore underlying ideas. It is to be noted that the approach implemented in this paper can be adapted for any kind of enclosure and in particular different types of waveguides. The channel capacity can be obtained in terms of the classical phase space density which is more advantageous as the DEA equivalent phase-space density is frequency insensitive and is also dependent on the surrounding physical environment of the system.

The calculations in this paper are motivated by the work of Miller in [3] and we aim to extend them to environments where multipath is important. Miller in his paper [3] has extensively discussed the possible communication channels between two volumes, one containing the transmitting antennas and the other receiving antennas, using physical models to estimate how good a connection we can make on a channel in free space. The paper concentrates on only analysing line-of-sight effects and does not incorporate multipath propagation. The main focus of the paper is on describing a mathematical formalism that incorporates multi-path interference effects. The approach implemented by Miller [4] allows one to define a unique set of optimal spatial channels for communicating between two arbitrary volumes. A sum rule is obtained in order to give quantitative information about the strength of the connection and the possible number of such channels.

The formalism discussed by Miller can also be adopted to interpret the maximum channel capacity in a MIMO setup inside an enclosure such as a waveguide. An approach similar to that implemented in [3] is presented in this paper where antennas within volumes are replaced by antennas on surfaces, such as patch antennas etc. The analysis of the transfer of information between the transmitting and receiving antennas within enclosures not only involves line-of-sight propagation, but also the multi-path propagation, which is crucial in MIMO communication. In the paper we see that the source function in the transmitting domain and the resulting wave in the receiving domain are expanded using complete sets of basis functions which we have adopted for surfaces in this paper. However, we do not address the issue of channel capacity maximization here as we mainly focus on modelling of wave propagation within an enclosure by a statistical model which is an extension of our previous work on wave propagation in freespace [5], [6], [7], [8], [9].

The propagation of correlation functions (CFs) and the use of Wigner distribution function (WDF) allows us to transform CFs of currents or emitted fields into functions of phase space co-ordinates as seen in [9]. We have also seen that the CF-based approach has proved an effective tool to predict radiation from printed circuit boards (PCBs) in [5]. Furthermore, the corresponding WDF formalism has proved effective in predicting the radiation pattern in free space, especially as it captures well the decay of the evanescent components in near field measurements. Placing such sources inside a cavity or enclosure [10], [11], [12] results in multi-path interference and hence makes the prediction of signal propagation more difficult. Thus, the relationship between the CF and the WDF will help predict these fluctuations in cavities reliably.

In the next few sections, we will first give a brief description of a waveguide and then move onto developing CF, understanding how the CF behaves for a model example and then get an insight into the dynamics of wave propagation in the phase space representation:-the WDF.

2. MIMO in a waveguide

In this section we mainly focus on understanding and visualising the MIMO setup within a waveguide. We use the concepts of transfer operator approach used in [4] to construct our CF for the case presented. The CF obtained is propagated using the propagator model developed in [13] along with the boundary conditions. The theoretical model developed in this paper will help predict channel capacity, number of channels needed to maximize information throughput etc. using the concepts of WDF. The method adopted here simplifies the problem of predicting channel capacity in terms of computational time and thus making it cost effective. This will enhance the future of chip architecture and information technology in a massive way. These models can be used in software packages upon reliably modelling EM field propagation in complex environments.

In [3], Miller has constructed a Hermitian kernel in terms of Green functions which is similar to the propagated CF discussed in [13]. The starting point for our discussion is to construct a channel matrix for a MIMO setup within an enclosure, where the antennas are mounted on plane surfaces, such as with patch antennas. The channel matrix is expressed in terms of a transfer operator which in turn is used to construct a Hermitian kernel, the Hermitian kernel is a propagated correlation function [9] from the source. The next logical step would be to exploit the relationship between the CF and the WDF. Based on the approach laid out in Miller's paper, we can obtain an approximation for the channel capacity in terms of the WDF. Before we proceed to the discussion on construction of CF, it is important first to understand what a waveguide is and later we define the correlation function, taking into consideration the boundary conditions.

A waveguide in electromagnetics and communication engineering refers to any linear structure that conveys information or electromagnetic waves between two endpoints. It is usually a hollow metal piece that carries radio waves. Such waveguides are used as transmission lines mostly at microwave frequencies, for purposes such as connecting microwave transmitters and receivers to their antennas, in devices such as microwave ovens, radar sets, satellite communications, microwave radio links etc. There are different types of waveguides for each type of wave. Here we consider a rectangular waveguide as shown in Fig. 1. The implementation of the model is done for a simple case of 1D waveguide. The propagation in a waveguide can be thought of as carried by rays travelling down the guide in a zig-zag path with repeated total reflection at the opposite walls of the waveguide. A propagation mode in a waveguide is one solution of the wave equation, or, in other words, the form of the wave. Due to the constraints of the boundary conditions, there are only limited forms for the wave function which can propagate in the waveguide. The lowest frequency in which a certain mode can propagate is the cutoff frequency of that mode. The mode with the lowest cutoff frequency [14] is the fundamental mode of the waveguide, and its cutoff frequency is the waveguide cutoff frequency.

Figure 1.

The top figure is a typical open rectangular waveguide where the direction of propagation is along the length of the waveguide. The figure in the bottom mimics the zig-zag propagation inside a waveguide where the ray undergoes total internal reflection at the walls of the waveguide.

Propagation modes are computed by solving the Helmholtz equation alongside a set of boundary conditions depending on the geometrical shape and materials bounding the region. The usual assumption for infinitely long uniform waveguides allows us to assume a propagating form for the wave, i.e. stating that every field component has a known dependency on the propagation direction (i.e. z). It is possible for many modes to propagate along a waveguide. The number of propagating modes in a waveguide increases with increasing frequency. In the next few sections we go on to consider the correlation function propagation for a rectangular 1D waveguide both in terms of the transfer operator and as well as in terms of the modal basis functions.

3. Transfer operator approach

In this section we make a concrete connection between DEA and full wave modelling using a transfer operator approach. The transfer operator approach proves to be a powerful tool for the application of semi-classical methods to complex quantum and wave problems. Especially in the case of cavities or enclosures, the transfer operator is closely related to boundary integral methods that provide exact numerical solutions. These semi-classical methods have found widespread application in problems with complex or chaotic ray limits. In the context of electromagnetic wave fields or vibro-acoustics, cavity problems with a variety of boundary conditions can be treated.

We here mainly focus on linear and stationary wave problems and implementation of the method for a MIMO system in a cavity. A transfer operator $\hat{T}$ is used to describe the wave problem under consideration. The transfer operator is defined on a $(d+1)$ dimensional space where d is the dimension of the surface under consideration. Here we consider a simple wave model;

$$-{\mathrm{\nabla }}^2\psi -k^2\psi =0.$$ (1)

Eq. (1) is the scalar Helmholtz equation in a d dimensional region Ω

The formulation of an operator on a boundary is given by the boundary integral equation. In [15] it is seen that the boundary integral equation serves as an application of a transfer operator to a wave function on the boundary, which refers to the Dirichlet boundary condition, or on its derivative, which refers to the Neumann boundary condition. The boundary integral equation gives a relationship between the wave function ψ and its normal derivative on the boundary. Now the solution to the homogeneous wave equation can be written in terms of the transfer operator in the form

$$\psi =\hat{T}\psi +{\psi }_0,$$ (2)

and Eq. (2) is referred to as the transfer operator approach [16]. Here ${\psi }_0$ is the source wave field that is incoming on the boundary, direct from the source and ψ denotes the trace of the wave fields on the boundary representing the components arriving at it. An explicit calculation of the transfer operator is not straightforward and has been mostly obtained for model systems [16], [17], [18], [19]. Bogomolny has obtained a general expression for the transfer operator using semi-classical approximation [15] when there exists a classical trajectory which connects the points x and ${\mathbf{\text{x}}}^{\prime }$ which corresponds to one Poincare′ mapping as shown in Fig. 2.

$$T(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })={\left(\frac{k}{2\pi i}\right)}^{(d)/2}\sum _{{\mathbf{\text{x}}}^{\prime }\to \mathbf{\text{x}}}D(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })\text{exp}(ikS(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })-i\mu \frac{\pi }{2}),$$ (3)

with

$$D(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })=\sqrt{\left|\text{det}\right(\frac{{\partial }^{2}S}{\partial \mathbf{\text{x}}\partial {\mathbf{\text{x}}}^{\prime }}\left)\right|},$$ (4)

where $S={\int }_{{\mathbf{\text{x}}}^{\prime }}^{\mathbf{\text{x}}}\mathbf{\text{p}}\dot{\text{d}\mathbf{\text{x}}}$ in Eq. (4) is the classical action along the trajectory and p is the classical momentum. The sum in Eq. (3) is over all the ray trajectories of one Poincare′ that connect two points. μ is the phase index and S can be complex if there is damping. In our example case of a Helmholtz equation there is only one trajectory from ${\mathbf{\text{x}}}^{\prime }\to \mathbf{\text{x}}$ for each pair of boundary points. In this special case we have $S(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })\equiv L(\mathbf{\text{x}},{\mathbf{\text{x}}}^{\prime })$, where $L_{\alpha }$ is the length of the chord from ${\mathbf{\text{x}}}^{\prime }$ to x.

Figure 2.

Schematic diagram of a two-dimensional system in coordinate space. Σ is the Poincare′ surface of section. x and ${\mathbf{\text{x}}}^{\prime }$ are two points on the Poincare′ surface of section connected by a classical trajectory of one Poincare′ map.

4. Correlation function in terms of transfer operator

Here we describe the physical problem along with expressing the channel matrix in terms of the modes and transfer operator which translates to matrix representation in terms of basis functions. Firstly, we consider two domains ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$ as input and output surfaces of the waveguide respectively, as shown in Fig. 3, on which the transmit and receive antenna arrays are mounted respectively. The signals from the transmitters may be denoted by $\varphi ({\mathbf{\text{x}}}_T)$. These generate waves $\psi ({\mathbf{\text{x}}}_R)$ at the receiver end, where ${\mathbf{\text{x}}}_T\in {\mathrm{\Omega }}_T$ and ${\mathbf{\text{x}}}_R\in {\mathrm{\Omega }}_R$. The channel matrix can be represented using different sets of basis functions which describe the source function in ${\mathrm{\Omega }}_T$ and the received signal in ${\mathrm{\Omega }}_R$. One way of representing the channel matrix is by considering an orthonormal modal basis set of functions that span the entire width of the waveguide on both the transmitting and receiving regions, namely, $u_1(\mathbf{\text{x}}),u_2(\mathbf{\text{x}}),....u_n(\mathbf{\text{x}}),...$, where $u_n(x)$ for 1-D case is given by here the boundary conditions assumed are

$$u_n(x)=\sqrt{\frac{2}{L}}\sin \left(\frac{n\pi x}{L}\right),$$ (5)

where L is the width of the waveguide. Here we assume the Dirichlet boundary conditions and that the boundaries of the waveguide are perfectly smooth which leads to total internal reflection. The assumptions made here typically simplify the modelling complexity that arises with other assumptions such as Robinhood boundary conditions etc. The variable n denotes the mode number and the modal basis is obtained for a simple geometry of a waveguide, which are sine types of functions.

Figure 3.

A MIMO set up inside a rectangular waveguide. The transmitting and the receiving antennas are restricted to regions Ω_T and Ω_R respectively. The signals are projected onto the space by the projection operators P_T and P_R.

We now want to express the channel matrix in terms of the transfer operator, which provides us with semi-classical approximations in order to obtain an expression in the context of MIMO communication. In this scenario, the transfer operator is given by,

$$T({\mathbf{\text{x}}}_R,{\mathbf{\text{x}}}_T)=\sum _{\beta }{D}_{\beta }({\mathbf{\text{x}}}_R,{\mathbf{\text{x}}}_T)e^{ik{L}_{\beta }}.$$ (6)

The transfer operator $\hat{T}$, connects the compact transmitter and receiver domains ${\mathrm{\Omega }}_T$, ${\mathrm{\Omega }}_R$. The term $L_{\beta }({\mathbf{\text{x}}}_R,{\mathbf{\text{x}}}_T)$ denotes the length of the path taken by a ray from the transmitters to receivers. The transfer operator can be visualized from Fig. 4 which shows multiple transmit and receive antennas and multi-path propagation along with line-of-sight propagation. Thus Eq. (6) represents the transfer matrix for the case considered here.

Figure 4.

The figure shows two transmitting and two receiving antennas on a surface with a scatter. α₁,..., α₄ denote the different paths a signal can take between the transmitters and receivers.

The transfer operator connects the two domains, ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$. However, the Eq. (6) can be equivalently written in terms of the modal basis function as

$$T({\mathbf{\text{x}}}_R,{\mathbf{\text{x}}}_T)=\sum _n{e}^{ikz\text{T}({\mathbf{\text{p}}}_n})u_n({\mathbf{\text{x}}}_R)u_n^{⁎}({\mathbf{\text{x}}}_T),$$ (7)

where k is the wavenumber and $\text{T}({\mathbf{\text{p}}}_n)$ is given by

$$\text{T}({\mathbf{\text{p}}}_n)=\sqrt{1-{\left(\frac{n\pi }{kL}\right)}^2},$$ (8)

where L is the width of the waveguide. The Eq. (7) is a model obtained in [9] which gives us a propagator rule that helps us analyse wave propagation in free space. The Eq. (7) and Rq. (8) help us account for the propagating modes and the evanescent components as the wave propagates away from the source.

The other set of basis functions we consider in our discussion is the finite dimensional basis of pixel functions which give a very simplified model of antennas. Let $w_1({\mathbf{\text{x}}}_T)$, $w_2({\mathbf{\text{x}}}_T),....w_i({\mathbf{\text{x}}}_T),...$ and $w_1({\mathbf{\text{x}}}_R),w_2({\mathbf{\text{x}}}_R),.....w_j({\mathbf{\text{x}}}_R),...$ be the pixel functions defined for the surfaces in the transmitting and receiving regions of the waveguide respectively, where $w_i({\mathbf{\text{x}}}_T)$ and $w_j({\mathbf{\text{x}}}_R)$ are given by

$$w_i({\mathbf{\text{x}}}_T)=\frac{1}{\sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_T|}}\{\begin{array}{cc}\hfill & 1,\text{in}{i}^{\text{th}}\text{pixel}\hfill \\ \hfill & 0,\text{outside}.\hfill \end{array}$$ (9)

$$w_j({\mathbf{\text{x}}}_R)=\frac{1}{\sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_R|}}\{\begin{array}{cc}\hfill & 1,\text{in}{j}^{\text{th}}\text{pixel}\hfill \\ \hfill & 0,\text{outside}.\hfill \end{array}$$ (10)

Here $i=1,..,N_T$ and $j=1,..,N_R$ represent the number of pixels on the transmitting and receiving regions and $|\mathrm{\Delta }{\mathbf{\text{x}}}_T|=\text{Length}({\mathrm{\Omega }}_T)/N_T$ and $|\mathrm{\Delta }{\mathbf{\text{x}}}_R|=\text{Length}({\mathrm{\Omega }}_R)/N_R$. The transmitting and receiving regions of the waveguide are filled in with these pixel type of functions. In the later part of the discussion we project these functions onto domains ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$ which are characterised by the modal basis functions. The above definition is interpreted as follows: At the i^th pixel on the transmitter end of the waveguide, the basis function $w_i({\mathbf{\text{x}}}_T)$ is 1 times the normalization factor for any ${\mathbf{\text{x}}}_T$ in the range of the pixel and similarly for the receiving end of the waveguide.

Now, that the physical problem and the different sets of basis functions are described, we move on to constructing the CF using the channel matrix information. The channel matrix H is approximated as a sum over all the paths between the transmitters and receivers. The first step towards constructing the channel matrix, for a complex scenario, is to construct it for the simplest basis set of pixel functions, wherein the functions are as defined in Eq. (9) and Eq. (10). We first describe the pixel representation of the channel matrix wherein the signals from the transmitting and receiving regions are in the form of step functions. If H is the channel matrix and the transmitting and receiving regions are pixelated as shown in Fig. 3, then the elements of the channel matrix can be defined in terms of the transfer operator $\hat{T}$ as

$${\mathbf{\text{H}}}_{ji}=\langle w_j|\hat{T}|w_i\rangle$$ (11)

$$=\int \text{d}{\mathbf{\text{x}}}_R\int \text{d}{\mathbf{\text{x}}}_T{w}_j({\mathbf{\text{x}}}_T)T({\mathbf{\text{x}}}_R,{\mathbf{\text{x}}}_T)w_i({\mathbf{\text{x}}}_T)\approx \int \text{d}{\mathbf{\text{x}}}_R\int \text{d}{\mathbf{\text{x}}}_T\frac{1}{\sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_R||\mathrm{\Delta }{\mathbf{\text{x}}}_T|}}T({\mathbf{\text{x}}}_R^j,{\mathbf{\text{x}}}_T^i)=\sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_R||\mathrm{\Delta }{\mathbf{\text{x}}}_T|}T({\mathbf{\text{x}}}_R^j,{\mathbf{\text{x}}}_T^i).$$ (12)

Here $T({\mathbf{\text{x}}}_R^j,{\mathbf{\text{x}}}_T^i)$ corresponds to the evaluation of T at the $i^{\text{th}}$ pixel at the transmitter end that connects to the $j^{\text{th}}$ pixel at the receiver end. This is a simple representation of the channel matrix where $w_i$'s and $w_j$'s are used as proxies for the patch antennas at positions i and j respectively. Thus, the channel matrix can be defined in two ways. One way of representing the channel matrix is by using the pixel basis function where we have considered the transmitters and receivers in the form of step functions (pixels) in their respective domains. This helps us actually obtain the upper bound for the channel capacity. That is, on increasing the number of pixels filling a given range, we observe that the channel capacity saturates, and on further increasing the number of pixels beyond some number, does not significantly increase the channel capacity, instead it saturates at a point. This in fact helps us to set the effective maximum number of transmitters and receivers at either end of a rectangular waveguide.

The second representation of the CF is through the use of the transfer operator and the projection operators (defined below), which is mostly used for analysis of channel capacity for MIMO communication in a waveguide. This operator representation can be analysed in matrix form by considering the basis functions for both the transmitting and receiving domains. We will see the different ways of defining the channel matrix later and present results for these various forms. In order to implement the second way of representation we again consider a 1D rectangular waveguide which is populated by transmitters and receivers at either ends of the waveguide covering a particular subset of the cross-section of the waveguide. ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$ represent the transmitter and receiver domains of the waveguide respectively. The idea here is that the domains ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$ may be populated arbitrarily densely with transmit and receive antennas, but that no antennas are placed outside these regions. The signal from the transmitters is projected onto the space by the projection operator ${\hat{P}}_T$ and the received signal is projected onto the space by a projection operator ${\hat{P}}_R$ as shown in Fig. 3. The direction of propagation of the signals is along coordinate z. We know $\hat{T}$ is the transfer operator that connects the two domains. We now define projection operators for the signals from both transmit and receive antennas, by

$${\hat{P}}_R:\psi ({\mathbf{\text{x}}}_R)\to \chi ({\mathbf{\text{x}}}_R)\psi ({\mathbf{\text{x}}}_R)$$ (13)

$${\hat{P}}_T:\varphi ({\mathbf{\text{x}}}_T)\to \chi ({\mathbf{\text{x}}}_T)\varphi ({\mathbf{\text{x}}}_T),$$ (14)

where $\chi ({\mathbf{\text{x}}}_R)$ and $\chi ({\mathbf{\text{x}}}_T)$ denote the characteristic function of ${\mathrm{\Omega }}_R$ and ${\mathrm{\Omega }}_T$ respectively. The functions $\psi ({\mathbf{\text{x}}}_R)$ and $\varphi ({\mathbf{\text{x}}}_T)$ can be expressed in a basis of eigenfunctions of a Hermitian kernel. The Hermitian kernel refers to the CF which we define in terms of the transfer and projection operators later. Before we define the channel matrix and the CF, it is necessary to establish a few preliminary definitions. We consider two sets of eigenfunctions, one for representing the transmitted signals and the other representing the received signals in their respective domains. Let ${\varphi }_1({\mathbf{\text{x}}}_T),{\varphi }_2({\mathbf{\text{x}}}_T),....{\varphi }_n({\mathbf{\text{x}}}_T)$ and ${\psi }_1({\mathbf{\text{x}}}_R),{\psi }_2({\mathbf{\text{x}}}_R),.....{\psi }_n({\mathbf{\text{x}}}_R)$ be the two sets of eigenfunctions of the Hermitian matrix ${\mathbf{\text{H}}}_{RT}^{\mathrm{†}}{\mathbf{\text{H}}}_{RT}$ and ${\mathbf{\text{H}}}_{RT}{\mathbf{\text{H}}}_{RT}^{\mathrm{†}}$ respectively, where ${\mathbf{\text{H}}}_{RT}^{\mathrm{†}}$ represents the Hermition conjugate of ${\mathbf{\text{H}}}_{RT}$, that are used to describe the signal $\varphi ({\mathbf{\text{x}}}_T)$ transmitted from the source and the signal $\psi ({\mathbf{\text{x}}}_R)$ that is received respectively and ${\mathbf{\text{x}}}_T\in {\mathrm{\Omega }}_T$ and ${\mathbf{\text{x}}}_R\in {\mathrm{\Omega }}_R$. Thus, these transmitted and the received signals can be expressed as

$$\varphi ({\mathbf{\text{x}}}_T)=\sum _m{b}_m{\varphi }_m({\mathbf{\text{x}}}_T)$$ (15)

$$\psi ({\mathbf{\text{x}}}_R)=\sum _m{d}_m{\psi }_m({\mathbf{\text{x}}}_R),$$ (16)

where $b_m$ and $d_m$ are constants. Defining the transmitted and received signals in terms of eigenfunctions of the respective domains, as in Eq. (15,16) forms the basis for our further discussions on CF and channel matrix.

From the definition of the transfer and projection operators, the channel matrix ${\mathbf{\text{H}}}_{RT}$ can be defined for the case where the transmitting and receiving antennas are restricted to their respective domains ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$ and taking into consideration that the signals from these regions are projected onto the space by the projection operators. Thus, ${\mathbf{\text{H}}}_{RT}$ which is a discrete representation alternatively of the operator ${\hat{H}}_{RT}$ is defined using Eq. (13-14) as

$${\hat{H}}_{RT}={\hat{P}}_R\hat{T}{\hat{P}}_T.$$ (17)

A special case of the operator ${\hat{H}}_{RT}$ is when the projection operators ${\hat{P}}_T$ and ${\hat{P}}_R$ are the identity operator $\hat{I}$, i.e., when ${\hat{P}}_T=\hat{I}={\hat{P}}_R$. This physically corresponds to the case where the entire width of the waveguide is populated by the transmitters and receivers. In this case we denote the channel matrix as H where the corresponding operator is $\hat{H}=\hat{T}$. Now that the channel matrix is projected on to the transmitting and receiving domain with the help of the projection and transfer operators, Eq. (17) can easily be expressed in terms of the modal basis functions defined in Eq. (5). In order to obtain the elements of the channel matrix ${\mathbf{\text{H}}}_{RT}$, we approximate the modal basis functions $u_n({\mathbf{\text{x}}}_T)$ at the transmitting region by taking an inner product with the pixel basis function which is approximated by

$$\langle w_i|u_n\rangle \approx \frac{1}{\sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_T}|}{u}_n({\mathbf{\text{x}}}_T^i).$$ (18)

Similarly the modal basis function for the receiving region can be approximated by performing an inner product with the pixel basis function as in Eq. (18). Thus, the matrix elements of ${\mathbf{\text{H}}}_{RT}$ can then be expressed in the bra-ket notation as

$${({\mathbf{\text{H}}}_{RT})}_{ji}=\langle w_j|{\hat{H}}_{RT}|w_i\rangle =\langle w_j|{\hat{P}}_R\hat{T}{\hat{P}}_T|w_i\rangle \approx \sqrt{|\mathrm{\Delta }{\mathbf{\text{x}}}_R||\mathrm{\Delta }{\mathbf{\text{x}}}_T|}\sum _n{e}^{-ikz\text{T}({\mathbf{\text{p}}}_n)}{u}_n({\mathbf{\text{x}}}_R^j)u_n^{⁎}({\mathbf{\text{x}}}_T^i).$$ (19)

Here the $u_n$'s are calculated from Eq. (5). Thus, the signal originating from the transmitter is projected onto space through the projection operator $\widehat{P_T}$, which is then transferred onto the receivers where the received signal is again projected onto the space through the use of the projection operator $\widehat{P_R}$. In our example of a waveguide, the channel matrix is numerically obtained using basis functions that are defined in terms of sine functions that are obtained by solving the Helmholtz equation with Dirichlet boundary condition. Thus, the channel matrix is obtained by summing over all modes.

Now that the various representations of the channel matrix have been discussed, the next step is to calculate the CF. As Miller has defined a Hermitian kernel in [3], and here we define an analogous Hermitian matrix, which is nothing but the CF in the current context using the channel matrix. Thus, the correlation function is obtained by calculating ${\mathbf{\text{H}}}_{RT}{\mathbf{\text{H}}}_{RT}^{\mathrm{†}}$. The corresponding operator notation is of the form ${\hat{H}}_{RT}{\hat{H}}_{RT}^{\mathrm{†}}$, where ${\hat{H}}_{RT}$ is defined in Eq. (19). Thus we expand the operator ${\hat{H}}_{RT}{\hat{H}}_{RT}^{\mathrm{†}}$ in terms of the transfer and projection operators as

$${\hat{H}}_{RT}{\hat{H}}_{RT}^{\mathrm{†}}=({\hat{P}}_R\hat{T}{\hat{P}}_T){({\hat{P}}_R\hat{T}{\hat{P}}_T)}^{\mathrm{†}}=({\hat{P}}_R\hat{T}{\hat{P}}_T).({\hat{P}}_T^{\mathrm{†}}{\hat{T}}^{\mathrm{†}}{\hat{P}}_R^{\mathrm{†}})={\hat{P}}_R\hat{T}({\hat{P}}_T{\hat{P}}_T^{\mathrm{†}}){\hat{T}}^{\mathrm{†}}{\hat{P}}_R^{\mathrm{†}}={\hat{P}}_R\hat{T}{\hat{P}}_T{\hat{T}}^{\mathrm{†}}{\hat{P}}_R.$$ (20)

Here $\hat{T}{\hat{P}}_T{\hat{T}}^{\mathrm{†}}$ is the propagated correlation function from the source. Similarly, we have

$${\hat{H}}_{RT}^{\mathrm{†}}{\hat{H}}_{RT}={\hat{P}}_T{\hat{T}}^{\mathrm{†}}{\hat{P}}_R\hat{T}{\hat{P}}_T.$$ (21)

It is to be noted that the definition in Eq. (21) is in terms of the operators which can also be equivalently expressed in terms of matrix representation as ${\mathbf{\text{H}}}_{RT}{\mathbf{\text{H}}}_{RT}^{\mathrm{†}}$ which is a projection of ${\mathbf{\text{P}}}_R{\mathbf{\text{T P}}}_T{\mathbf{\text{T}}}^{\mathrm{†}}{\mathbf{\text{P}}}_R$ onto a pixel basis as in Eq. (11,12), where ${\mathbf{\text{P}}}_R$ and ${\mathbf{\text{P}}}_T$ are the corresponding matrix representation of the operators ${\hat{P}}_R$ and ${\hat{P}}_T$ respectively.

Thus we have obtained the Hermitian kernel in both operator and matrix notation, which will be the starting point for our further discussions.

5. Correlation function

In the previous section, we have extensively presented discussion on obtaining the CF. We have shown that the initial CF can be propagated from the transmitter end, within a waveguide, to obtain the received CF. In this section we present the initial and propagated CF for a particular example of a waveguide. The CF obtained here is again for a 1D rectangular waveguide of width 0.5m. The frequency of operation is 3GHz, which means the wavelength $\lambda =0.1$m. The width of the waveguide is 5 times that of the wavelength. Thus, there are nearly 10 propagating modes that one observes as we move away from the source. We have used 100 transmitters and receivers in each of the transmitting and receiving domains for illustration purposes. We first present the result for the case where the entire width of the waveguide is populated by antennas, that is, ${\hat{P}}_T=\hat{I}$. Fig. 5 shows the source CF, for $z=0.005$m on the top left hand corner in which we see that the entire diagonal shows a red strip that represents the signals from antennas that are spanning the entire width of the waveguide. The propagated CF is obtained by first calculating ${\mathbf{\text{H}}}_{RT}$ using Eq. (19) and then evaluating ${\mathbf{\text{H}}}_{RT}{\mathbf{\text{H}}}_{RT}^{\mathrm{†}}$ which are shown for very small and large values of z. The small values of z is chosen to represent the CF distribution very close to the source which are characterized by evanescent contributions which will be clearly evident from the corresponding WDFs. On the top right hand corner and the bottom row in Fig. 5 show the propagated CF away from the source for small z. The CF at $z=0.005$ m swells as we move to larger heights $z=0.01$ m, $z=0.02$ m, and $z=0.05$ m. This corresponds to an increase in the correlation length and thus serves as evidence for the presence of evanescent contributions. As we move from $z=0.02$m to $z=0.05$m, we do not observe any further significant evolution in CF distributions. We observe the non-evolution even for larger values of z which are not shown here in order to avoid monotony. This implies that all the evanescent components have decayed and we are only left with the propagating regime at a distance $z\gtrsim \lambda$ away from the source. It is also important to know that, ${\hat{P}}_R=\hat{I}={\hat{P}}_T$ which implies that $\hat{H}{\hat{H}}^{\mathrm{†}}=\hat{T}{\hat{T}}^{\mathrm{†}}$. For a better understanding we present the WDF plots for these CFs in the later subsection on Wigner functions.

Figure 5.

Theoretical CFs are shown at the top and bottom for a frequency of 3 GHz at different heights with source at z = 0.005m (top left-hand corner) and propagated to heights z = 0.01m (top right-hand corner), z = 0.02m (bottom left-hand corner) and z = 0.05m (bottom right-hand corner).

Now we will see how the CF distribution changes for the second scenario where a proper subset of the waveguide cross section is excited with transmitters and receivers. Here the transmitting and receiving regions are restricted to domains ${\mathrm{\Omega }}_T$ and ${\mathrm{\Omega }}_R$. The CF obtained using the definition in Eq. (20), where ${\hat{P}}_T\ne \hat{I}\ne {\hat{P}}_R$. Here the region of excitation is restricted between 0.2m and 0.4m of the width of the waveguide. Hence in the CF plots in Fig. 6 we see the source CF ($z=0.005$m) on the top left hand corner. In this plot we observe that only a part of the diagonal shows a red stripe corresponding to the signals from antennas, which is as expected. Again, as we move away from the source, it is observed that for very small values of z the stripe expands, which is similar to the observation made in Fig. 5. This as already stated earlier corresponds to the increase in correlation length which in turn corresponds to the decay of the evanescent contributions. The stripe expands until $z=0.05$m after which, at $z=0.15$m and $z=0.2$m the region along the diagonal expands and contracts. It forms an oscillating pattern as we propagate the source CF, which can be observed from the last row of Fig. 6. The corresponding WDF will give a better insight on these observations.

Figure 6.

Theoretical CFs are shown at the top, middle and bottom rows for a frequency of 3 GHz at different heights with source at z = 0.005m (top left-hand corner) and propagated to heights z = 0.01m (top right-hand corner), z = 0.02m, z = 0.05m (middle row) and z = 0.15m, z = 0.2m (bottom row).

6. Wigner function

This section is dedicated to analysing the WDF representation of the CF discussed previously. In this section we analyse the propagation of waves within a confined space, such as, a waveguide. This section also provides the complexity of the distribution and the presence of evanescent contributions very close to the source. In this section the WDF of the CFs discussed in the previous section is presented for a systematic analysis. The WDF which describes the position in terms of the vector x and the direction in terms of the momentum vector p which is obtained by performing Fourier transform and is given by

$$W_z(\mathbf{\text{x}},\mathbf{\text{p}})={\left(\frac{k}{2\pi }\right)}^d\int e^{ik\mathbf{\text{x}}\cdot \mathbf{\text{q}}}{\mathbf{\text{H}}}_{RT}{\mathbf{\text{H}}}_{RT}^{\mathrm{†}}d\mathbf{\text{q}},$$ (22)

where $\mathbf{\text{x}}=({\mathbf{\text{x}}}_1+{\mathbf{\text{x}}}_2)/2$, d is the dimension of the transverse space and k is the wave-number.

The WDFs for the corresponding CFs in Fig. 5 and Fig. 6 can be obtained by performing a simple coordinate rotation [9] and then performing a Fourier transform with respect to the displacement variable as in Eq. (22). In this section we show the WDF plots for both scenarios as discussed in the previous section while discussing CFs. Fig. 7 shows the source WDF, i.e. for $z=0.005$m on the left of the top row. From this plot we observe that the WDF very close to the source, extends beyond the region $-1<|\mathbf{\text{p}}|<1$ which corresponds to the evanescent components. The signals from the sources are rapidly oscillating and the wavelength is greater than the distance between the antennas at the source, which plays a role in the generation of evanescent waves at the source or very close to the source. From the plot of the WDF for $z=0.005$m we can also observe that the transmit and receive antennas span the entire width of the waveguide, which is 0.5m in this illustration, as expected. As we move away from the source, we observe that the evanescent components decay, as shown at the right side of the top row in Fig. 7. Once the evanescent components have fully decayed we observe that only the propagating modes remain, which is evident from the bottom row of Fig. 7. We have observed that there are 10 propagating modes, which is as expected from the expression $L=\frac{q\lambda }{2}$, where q gives the number of nodes when L (0.5m) and λ (0.1m) are known. As we propagate away from the source, we observe that the WDF starts to rotate, particularly as observed in free space. Since the waveguide is hollow and the region inside the waveguide is assumed to be free space, we would expect the waves to propagate as in free space for small values of z, until the rays reach the walls of the waveguide. From Fig. 7 we observe very slight shearing effect, though it is not evident at once from the plots.

Figure 7.

Theoretically obtained WFs are shown along the two rows for a frequency of 3 GHz at different heights with source at z = 0.005m (top left-hand corner) and propagated to heights z = 0.01m (top right-hand corner) and z = 0.02m, z = 0.05m (bottom row) with transmitters and receivers spanning across the width of the waveguide.

In order to visualize and understand why there isn't much shearing or rotation we turn to the phase space density representation as shown in Fig. 8. The source density ${\rho }_0$ for this scenario would be a rectangular patch that would cover the entire region of phase space. Thus, we replace the WDF plot for $z=0$ by a rectangular patch on which the density is constant. Again, as we propagate in free space we expect an S-shaped curve as in Fig. 8, but inside a waveguide the tail ends of the S-shaped curve gets folded inside on either ends so that they lie within the region occupied by the waveguide. This is the mechanism of propagation within the waveguide in terms of WDF or phase space representations. Since, the phase space density covers the entire region, the tail ends which would lie outside the region of the waveguide in the absence of the walls would be comparatively less. Thus, only a small part of the curve needs to get folded so that it lies within the region of the waveguide and the region of intersection of the S-shaped curve coincides with the entire rectangular patch at the source. Hence, in the WDF plots we do not observe much of these effects of the structure for various values of z. With this understanding, we move onto the second scenario.

Figure 8.

The figures shows the phase space density at z = 0 representing the rectangular patch and free space propagation represented by the S-shaped curve for both the domains Ω_T and Ω_R. The phase space density in Ω_T is the pre-image of the phase space density in Ω_R. The shaded region in both the left and right plot corresponds to the region of intersection.

In the second scenario we restrict the region of excitation of the antennas as shown in Fig. 9. The source WDF on the left of the top row in Fig. 9 shows similar structures as the first scenario discussed previously, except that the region here is restricted between 0.2m and 0.4m. The region between 0.2m and 0.4m is chosen for illustrative purpose only. Any region within the width of the waveguide can be chosen as per the requirement of the problem. This will act as a smaller waveguide except that the walls are still at 0 and 0.5m. Again the propagating regime lies in $|\mathbf{\text{p}}|<1$ and we observe evanescent components for the source WDF similar to the previous discussion. As we move away from the source we observe the rapid decay of the evanescent components and we are just left with the propagating part, which is evident from the right hand plot of the top row and the plots in the middle and bottom row of Fig. 9. We observed from Fig. 6 that there are 4 propagating modes which is again as expected as the width L is 0.2m with $\lambda =0.1$m. The other important observation from these plots is that the shearing effect is more evident in each of them, contrary to what we observed in the previous scenario. We also observe that the WDF rotates as we move further away from the source. In order to understand these observations we again take the help of the phase space density representation.

Figure 9.

Theoretically obtained WDFs are shown along the four rows for a frequency of 3 GHz at different heights with source at z = 0.005m (top left-hand corner) and propagated to heights z = 0.01m (top right-hand corner), z = 0.02m, z = 0.05m (second row), z = 0.1m, z = 0.15m (third row) and z = 0.2m, z = 0.25m (fourth row) with transmitters and receivers spanning across a small region of the waveguide.

In the Fig. 8, we see that the S-shaped curve coincided with the source rectangular patch leaving only a small portion of the tail ends outside the region of intersection, which is unlike the present case where only a small region of the S-shaped coincides with the rectangular patch, leaving the larger part of the propagated region outside the rectangular patch if there were no walls. The present scenario is more like what we observe in Fig. 9. We can make similar observations as made previously where the region that lies outside the rectangular patch gets folded back into the rectangular patch (region of the wavegyuide where the antennas are excited) at the source. In Fig. 9, the phase space density is restricted to a smaller region at the source and hence the ray density has to travel for a longer distance in free space in order to reach the walls of the waveguide, but, on encountering a wall the tail ends get folded which corresponds to the rotation in the WDF plots shown in the middle and bottom rows. Thus, the rotation is due to the total internal reflection of the rays at the two walls of the waveguide. It is now easier to visualize that folding of one tail end of the S-shaped phase space density corresponds to the reflection of the rays at one wall and the folding of the other tail end to correspond to the reflection of the rays at the other wall of the waveguide. On folding these tail ends would finally result in a complex structure in the WF representation as evident from the plots in the bottom row. The expansion and contraction of the blobs in the CF plots in Fig. 6 corresponds to the rotation of the WDF in Fig. 9 which in turn corresponds to the total internal reflection of the rays. The complexity of the WDF increases with the increase in the number of reflections the rays undergo as we propagate further away from the source.

The WDF gives minute details about wave propagation, structural details, evanescent components, modal information etc. and hence the complexity. In the case of phase space density, the analysis is more intuitive but at the same time captures the qualitative nature of the WDF which makes it computationally more effective in more realistic situations. In most situations the CF measurements are not feasible due to a number of factors such as the fact that complex sources are extremely chaotic in the context of electromagnetic compatibility (EMC) and a near field measurement in particular within enclosures would be very difficult. The effects such as interference, diffraction, noise etc. make it even more challenging to experimentally obtain measurements. In such scenarios it is best to use DEA method to predict the nature of propagation for a given set of conditions.

7. Conclusion

In this paper we have discussed how wave propagation within enclosures can be analysed by visualizing it through the CF and the WDF. We have shown that it serves as one of the numerical computational technique that works effectively in comparison to traditional techniques such as FEM, FDTD etc. We have also introduced the wave transfer operator approach that plays an important role in modelling a MIMO setup within an enclosure where the transmitting and receiving antennas are restricted to their domains. In this paper we have extended Miller's theory [3] on MIMO communication between volumes in free space to communication between surfaces within enclosures. In particular, we have established a relationship between the channel matrix stated in Miller's paper to our statistical approach that exploits the relationship between the field-field CF and the WDF. We have analysed the CF and the WDF representations for a simple 1D rectangular waveguide, which can be extended to analyse dynamics of wave propagation within any enclosure such as reverberant chambers etc. WDF plays an important role in the analysis of evanescent waves which are not incorporated in the classical phase space density. In this paper we have achieved our main goal to extend the free space propagator model to analyse emissions within enclosures by solving the Helmholtz equation with Dirichlet boundary conditions.

CRediT authorship contribution statement

Deepthee Madenoor Ramapriya: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Kaliprasad C S: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Hemanthkumar C: Conceived and designed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.

Declaration of Competing Interest

The authors declare no competing interests.

Data availability

The data that has been used is confidential.