To investigate the Temporal and Spatial correlation behavioral characteristics of MIMO fading channels

5895 | P a g e M a y 1 5 , 2 0 1 5 To investigate the Temporal and Spatial correlation behavioral characteristics of MIMO fading channels Rachna Mahey, Jyoteesh Malhotra ECE Department, Guru Nanak Dev University, Regional campus, Jalandhar, INDIA rachnamaheyr@gmail.com ECE Department, Guru Nanak Dev University, Regional campus, Jalandhar, INDIA jyoteesh@gmail.com ABSTRACT This paper deals with the analysis of performance bounds of narrowband multiple-input multiple-output (MIMO) fading channels considering the transmission links from the mobile station (MS) to the base station (BS) through non-line-of-sight (NLOS) propagation conditions. It is assumed that an infinite number of local scatterers surround the MS and the BS. The Temporal as well as Spatial correlation properties of channel models are explored along with their capacity. The effect of number of scatterers; Doppler frequency; and isotropic/non-isotropic scattering conditions, on MIMO channel models is investigated in terms of statistical properties of one ring and two ring models, i.e. Time Autocorrelation Function (ACF) and two-dimensional (2D) Space Cross-correlation Function (CCF).


INTRODUCTION
The propagation conditions, determine the channel characteristics, and the channel capacity of a multiple-input multipleoutput (MIMO) system, thus it is important to develop MIMO channel models, to investigate the impact of the channel parameters on the capacity of MIMO systems in various propagation environments [1]. The appropriate MIMO fading channel models can help in reducing the costs of research and development, by simplifying the required computations and measurements on the design under different scenarios; and hence the system performances can be predicted accurately. Therefore, the realistic models of mobile fading channel are of great importance for designing, analysis, optimization, and testing of wireless communication systems [2]. A sophisticated MIMO channel models are required [3] to develop advanced mobile communication systems employing multiple antennas at the transmitter (MT) and receiver side (MR), that precisely reflect all the relevant features of physical channel. For designing and testing of MIMO Mobile-to-Mobile (M2M) systems, the Geometrically-Based Stochastic Model (GBSM) [4] is applied for narrowband MIMO channels. The most well-known narrowband MIMO channel models are; the one-ring model and the two-ring model. In M2M communication channels [5], the transmitter and receiver both are in motion; means one vehicle moving in a city tries to communicate with other moving vehicle at other location; for example police cars, emergency vehicles etc. Thus, the Lineof-Sight (LOS) is obstructed by the buildings or any other type of obstacle (may be scatterers) between the transmitter and receiver. The One-Ring Model assumes that the BS (transmitter) is not obstructed by any local scatterers, while the MS (receiver) is located in the center of a ring of surrounding scatterers [6]. This sort of assumption is suitable for propagation environments in the rural and sub-urban areas, where tall BS antennas are used. The Two-Ring Model assumes that both the transmitter and receiver are surrounded by scatterers [6], as in indoor channels.
In the literature, several models exist that allow for an appropriate simulation of mobile channels. Oestges, C. and Clerckx, B. [4], illustrate that an important aspect of MIMO channel modeling consists of describing the correlations between pairs of transmitter and receiver antennas. For Geometry-based models [7], instead of modeling the environment precisely, the scatterers having complex Gaussian properties are specified by a spatial distribution. Parizi, F. Z. and Mehrjoo, M. [8] presents a MIMO geometry-based stochastic channel models (GSCM) for regular shaped scattering environments, such as, one-ring and two-ring models. Avazov, N. and Patzold, M. [9], investigates the geometrical one-ring scattering model assuming single-bounce scattering in LOS and NLOS propagation environments. Analytical solutions are analyzed for the space-time-frequency CCF (STF-CCF), the 2D space CCF, the time-frequency CCF (TF-CCF), the temporal ACF, and the frequency correlation function (FCF). The comparison results [8], [9] reveal an excellent agreement between theory and measurements, which confirms that the derived reference model is valid. Wang, S. and Abdi, A. [10], also proposes a single-bounce two-ring statistical model for time-varying MIMO flat fading channels, that provides a useful channel characterization required for efficient design and performance prediction of multi-antenna systems. Talha, B. and Patzold, M. [11], derive a stochastic narrowband MIMO reference channel model from the geometrical three-ring scattering model. The general analytical solutions are found for the four-dimensional (4D) space-time CCF, 3D spatial CCF and the temporal ACF, which show that various CCFs describing the simulation model closely approximate the corresponding CCFs of the reference model. Cheng, X. and Laurenson, D. I. [12], extends the narrowband one-ring MIMO model to a new wideband multi-ring model for macro-cell scenarios. Patzold, M. [13] shows that the parameters of the simulation model can be determined by using the EMEDS (Extended Method of Exact Doppler Spread) and LPNM (Lp-norm method). The application of the former method is highly recommended for the isotropic scattering scenarios, while the latter exhibits its full capabilities when the AOD and AOA are non-uniformly distributed, as it is generally the case in real-world environments (non-isotropic scattering). This paper contributes to the one ring and two-ring models, by focusing on their performance analysis under different channel conditions. Stochastic simulation models for MIMO channels under the assumption of isotropic and non-isotropic scattering are investigated. The starting point of the performance evaluation is a non-realizable stochastic MIMO reference model as it has been developed for the one ring and two-ring scattering models in [6] [11]. General analytical solutions are provided for the 2D space CCF, time autocorrelation function (ACF), and Capacity distribution [3]. To compute the parameters of the simulation model, two methods are proposed [6] [14]. The first method is an extension of the MEDS, i.e. extended method of exact Doppler spread (EMEDS), which provides the closed-form expressions for the model parameters. EMEDS is developed for propagation scenarios described by uniformly distributed angles of departures (AOD) and angles of arrivals (AOA). The second method is Lp-norm method (LPNM) which provides the model parameters for many types of distribution functions of AOD and AOA; as the Von-Mises distribution [2] and the Gaussian distribution, however LPNM is more complex. In this paper, the Temporal and Spatial correlation behavioral characteristics of MIMO fading channels are analyzed and the effect of number of scatterers; Doppler frequency; and Isotropic/Nonisotropic scattering conditions, on MIMO channel models is investigated. The proposed methods can provide an important framework for system designers of advanced mobile communications, to verify new transmission concepts employing MIMO techniques under realistic propagation conditions. The rest of this paper is organized as: The geometrical, stochastic reference, stochastic simulation and deterministic simulation models for geometrical one-ring and two ring models, is described briefly in Section 2. The simulation methodology and environment is illustrated in Section 3. The performance results are evaluated in Section 4.

DETERMINISTIC CHANNEL MODELING
The simulation models are based on the principle of Deterministic channel modeling [6] [13]. Starting from the geometrical one-ring (or two ring) model, a stochastic and a deterministic simulation model is derived via a non-realizable stochastic reference model. The generalized concept of deterministic channel modeling used in this section is described by Figure 1: M a y 1 5 , 2 0 1 5

The One-Ring MIMO Channel Model:
The One-ring model is preferred in modeling sub-urban environments [3], where the BS and MS are considerably distant from each other and the BS is at much higher height than the MS. Thus, it is assumed that the BS (transmitter) has little or no scatterers surrounding it, as may be placed on a very high point, for e.g. a hill, mountain or Tower; while the MS (receiver) is placed at a much lower point and is assumed to be surrounded by many scatterers existing on a ring around the MS [8]. In Mobile-to-Mobile (M2M) communication systems, an example of this model may be an airplane or helicopter communicating with a ground vehicle. In this subsection, modeling, analysis, and simulation of one ring model are described briefly.

Geometrical One Ring Scattering Model:
The geometrical one ring model for MIMO channel employs multi-element antenna arrays. The Figure 2 contains description of all components of the model with local scatterers around the MS. It is assumed that the LOS path between the BS and the MS is obstructed. Here, D is the distance between MS and BS; R is radius of the ring of scatterers, a T max denotes the maximum AOD (angle of departure) observed from the BS; a n R is the AOA (angle of arrival) at MS; β T is the BS's tilt angle (in degree); β R is the MS's tilt angle; d T is antenna inter-element spacing at BS; d R is antenna element spacing at mobile station; a T max is one half of the maximum AOD observed at the BS; and a v is the MS's angle of motion [3] [6].

Reference Model for the One-Ring MIMO Channel Model:
For One-ring channel model, MT x MR MIMO channel with local scatters lying on a ring around the MS is assumed as a reference model. It is assumed that the number of local scatterers is infinite, i.e. (N= ∞). The complex channel gain hkl (t) of the reference model describing the link from the lth transmitter antenna element A T l (l = 1, 2, . . . , MT) to the k th receiver antenna element A R k (k = 1, 2, . . . , MR) is expressed as [13]: (1) M a y 1 5 , 2 0 1 5 Here θn is the phase change due to the interaction with the nth scatterer Sn. The time ACF of the complex channel gain hkl (t) derived from the space-time CCF is obtained as [6]: Here * denotes the complex conjugate operator. The 2D space CCF is given as [13]: Also, the instantaneous channel capacity of one ring channel model C(t) is defined as [13]:

Simulation Models for the One-Ring MIMO Channel Model:
An efficient simulation model is obtained from given reference model derived from the plane wave model by applying the generalized concept of deterministic channel modeling. In the third step, a stochastic simulation model is derived from the reference model simply by replacing the infinite number of scatterers by a finite number N. The fourth step is the step from the stochastic simulation model to the corresponding deterministic simulation model, which is performed by fixing all model parameters, including the phases.

Stochastic Simulation Model:
The time ACF rˆhkl (τ) of the complex channel gain hˆkl (t) of the stochastic simulation model is expressed as [6]: Finally, the 2D space CCF ρˆ(δT , δR) of the stochastic simulation model is derived as follows [6]: Deterministic Simulation Model: The practical realization of a stochastic simulation model requires theoretically the simulation of an infinite number of sample functions. The time ACF of the deterministic process h˜kl (t ) is obtained as [6]: The 2D space CCF of the deterministic simulation system, which is derived as follows [6]:

Parameter Computation Methods (isotropic and non-isotropic):
The objective of any parameter computation method is to find proper values for model parameters, such that the statistical properties of the deterministic (stochastic) simulation model are sufficiently close to those of the stochastic reference model. The model parameters to be determined are the discrete AOAs α R n (n = 1, 2, . . , N).

Extended method of exact Doppler spread (EMEDS):
Isotropic scattering around the transmitter is characterized by a uniform distribution of the AOD α T . For isotropic scattering around the receiver, we employ the EMEDS to determine the set of discrete AOAs {α R n}Nn=1. This method is a special case of the generalized method of exact Doppler spread (GMEDSq). According to the EMEDS, the model parameters α R n are determined by [6]: Where, α R 0 is called the angle-of-rotation.

Lp-norm method (LPNM):
To describe non-isotropic scattering scenarios, it is assumed that the AOD αT follows the von Mises distribution, for describing the measured data [2]. For non-isotropic scattering around the receiver, the LPNM can be applied. The application of this method to the present problem requires the minimization of the two Lp-norms [14]:

Two-Ring MIMO Channel Model
The Two-ring model is similar to the One-ring model; the difference is in the vertical position of the BS, which is not as higher as that in One-ring model. So, it is assumed that the BS is also surrounded by the scatterers existing on a ring along with the MS. Therefore, the model has two rings, one surrounding the BS (which is the transmitter); and the other surrounding the MS (which is the receiver). In Mobile-to-Mobile communications, the two ring model may be seen as a helicopter flying low in the city communicating with a ground vehicle.

Geometrical Two-Ring Scattering Model:
In this subsection, the geometrical two-ring scattering model for a narrowband MIMO channel is described briefly as illustrated in Figure 3. In the two-ring model, only local scattering is considered, because it is assumed that the contributions of remote scatterers to the total received power can be neglected due to high path loss. For simplicity, it is assumed that both transmitter and receiver are equipped with only two Omni-directional antennas.

Simulation Models for the Two-Ring MIMO Channel Model:
This subsubsection presents a stochastic as well as a deterministic simulation model by applying the generalized principle of deterministic channel modeling.

Parameter Computation Methods:
In this section, two methods for the computation of the parameters determining the statistics of the MIMO channel simulation model are represented. The first method is the EMEDS [9], which is highly recommended in case of isotropic scattering. The other methods is LPNM [11] which can be applied to any given distribution of the AOD (AOA), including the von Mises distribution [2], the Gaussian distribution, the truncated Laplacian distribution, and the truncated uniform distribution [6].

SIMULATION METHODOLOGY AND ENVIRONMENT
In this paper, the simulation models are based on the principle of Deterministic channel modeling which consists of the steps shown in Figure 1. The simulation of the one ring and two ring MIMO channel models is divided into two parts. First part deals with the computation of the AODs at the transmit side, and the other part determines the AOAs where each one is computed individually. By combining these two results the channel gains as well as the impulse response which characterizes the behavior of channel model are determined. The different steps for the simulations are: 1. The AOD and AOA are computed using two fundamental methods, by simply choosing the right parameters. The autocorrelation function of the simulation model is taken as criteria to evaluate the performance of each computation method.
2. In the second step, the channel gains are computed by using the obtained parameters for multiple uncorrelated fading waveforms.
3. The option PLOT is used to sketch the performance criteria, such as temporal ACF and space-time CCF.
4. The capacity of simulated channel model is presented using ecdf (empirical cumulative distribution function).
The resulting MIMO channel simulations takes not only the spatial correlation properties at the transmit side and the receive side into account, but models also accurately the Doppler Effect. The different parameter computation methods used are EMEDS and LPNM method. For each parameter computation method, the time ACF and 2D space CCF of the simulation models are depicted as proper statistical quantities. The simulations are based on the isotropic scattering assumption, i.e., p(φT) = p(φR) = 1/(2π) and non-isotropic assumption. The tilt angles are defined as α T = α R = π/2, and the angle of motion αv is set to π. The ring radii RT and RR were equal to 10 m. A maximum Doppler frequency of fmax = 91 Hz and 200 Hz is assumed, and the wave length is set to λ = 0.15 m. The parameters φ(m) T and φ(n) R of the stochastic simulation model have been computed by using the proposed extended MEDS with M = 20 and N = 40. The parameters used are as shown in Table 1.

RESULTS AND DISCUSSIONS
The time ACF and 2D space CCF are considered to compute the performance bounds of MIMO channel models by varying the number of scatterers; Doppler frequency; and isotropic/non-isotropic scattering conditions. The simulation results for different parameters values coming from the EMEDS method and the LPNM method (for isotopic and nonisotropic scattering respectively) are illustrated as: The Isotropic Scattering is investigated by choosing κ = 0 and mα = 0 and the method used for computing simulation parameters is EMED. Figure 4(a) and 4(b) shows the time ACF and 2D space CCF ρˆT(δ T, δ R ) for one ring channel model under isotropic scattering conditions. For the simulation model, an appropriate number of discrete scatterers N located on the rings around the source MS is selected (N=20). The simulation parameters chosen are; N= 20, β BS = 90, β MS = 90, α T max = 2, f max = 91Hz, κ = 0, m α = 0, α v =180 and p = 2. The time ACF is obtained for reference and simulation model, which shows the approximation rh kl (τ) ≈ r˜h kl (τ) is excellent over the interval from 0 to τmax. However, for τ > τ max , the time ACF's are inconsistent. The space CCF of simulation model, ρ˜(δ T ,δ R ) is evaluated using the discrete AOAs α R n obtained by applying the EMED.  Figure  5(a) shows that the time ACF inconsistencies are increased for τ > τ max on increasing the number of scatterers. It can be observed from Figure 5(b) that the space CCF decreases by increasing the number of scatterers. However, the decay of the 2D space CCF ρ 11,22 (δ T ,δ R ) is faster along the δR direction. M a y 1 5 , 2 0 1 5  (Figure 4(a) and 4(b)). Moreover, from Figure 6(b), it can be seen that the space CCF is highly correlated over a large range of antenna element spacing's δ T and δ R . This can be concluded from the fact that even for large antenna element spacing's, for example, δ T = δ R = 3λ, the absolute value of the 2D space CCF |ρ 11,22 (δ T ,δ R )| equals approximately one half of its maximum value. M a y 1 5 , 2 0 1 5 For Non-isotropic Scattering, a high-performance parameter computation method LPNM is used for computing the AOD (angle of departure) as well as AOA (angle of arrival), since the EMEDS cannot be used to determine the model parameters α R n for non-isotropic scattering. Figure 7(a) shows the time ACF ρT(δT, δR) of reference and ρˆT(δT, δR) simulation models; and Figure 7 . This explains the discrepancy between rhkl (τ) and r˜hkl (τ) if τ > τmax.    Figure 10(a), it can be seen that a capacity of 2bps/Hz (for example) can be acquired with the probability of 0.8. Also, a capacity of 1.35bps/Hz is equi-probable; means cumulative probability is 0.5. Figure 10(b) shows that capacity of 2.2bps/Hz can be obtained with probability of 0.5.  Figure 11(a). Also from Figure 11(b), it can be seen that the space CCFs are highly correlated over a large range of antenna element spacing's.  Figure 12(a), it can be observed that the 2D space CCF decreases as the antenna inter-element spacing is increased. Figure 12(b) shows that under isotropic scattering conditions, the close mapping of the models is obtained, with the increase in number of scatters N.   By analyzing these figures, the simulation results for different parameter values coming from the EMEDS method for isotropic scattering and the LPNM method for non-isotropic are found. For each parameter computation method, the ACF and CCF of the simulation model are acquired. By increasing the number of scatterers, the Space CCF ρ 11,22 (δ T ,δ R ) decreases quickly along the δR direction. However, the Time ACF and Space CCF remain the same by increasing the Doppler requency. On comparing the CCF of above figures, it shows that best performance is given in simulation results for non-isotropic scattering model, as compared to the isotropic scattering model. Reason behind this better performance is LPNM methods given best fitting parameter values for non-isotropic scatter model. The LPNM can be applied for minimizing the error function of the one-ring model parameters and the two-ring model parameters. Thus, LPNM is equally beneficial for the computation of the channel model parameters under non-isotropic scattering conditions. Under isotropic scattering conditions, a very good fitting is obtained until δR/λ=N/4 and δT/λ =N/4. With the increase of number of scatters N, the close mapping of the models is obtained. Therefore, an efficient adaptive modulation techniques are required.

CONCLUSION
A profound knowledge of mobile fading channels is required for the development, evaluation, and assessment at practical conditions of present and future wireless communication systems. Hence, the modeling, analysis, and simulation of MIMO fading channels are important segment for the initiation of mobile communications. This paper deals with the performance analysis and simulation of MIMO fading channels for mobile communication systems using one ring and two ring models. The Temporal and Spatial correlation behavioral characteristics of MIMO fading channels are investigated using One-Ring and Two-Ring models. For one-ring model, it is assumed that only the transmitter, which is the MS, is surrounded by an infinite number of local scatterers. Whereas, for the two-ring models, it is assumed that the scattering objects are located near both the transmitter (BS) and the receiver (MS). A general procedure is presented for the design of stochastic simulation models for frequency-nonselective MIMO fading channels based on the geometrical one ring and two-ring scattering models under different channel conditions.
Two methods are used for computing the parameters of the simulation model; i.e. the extended MEDS (EMEDS) for isotropic scattering and a variant of the Lp-norm (LPNM) method for non-isotropic scattering. A performance investigation of the LPNM revealed that this method outperforms the EMEDS slightly in case of isotropic scattering. The system performance decreases if the spatial correlation is increased. With increasing the number of scatterers, the performance of channel models degrades, thus efficient adaptive modulation techniques are required for better performance. Besides, the MIMO channel simulations enable an exact analysis of the channel capacity by simulation. The proposed procedure for the design of MIMO channels provides a fundamental framework for the test, optimization, design, and analysis of multielement antenna communication systems.