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International Journal of Antennas and Propagation
Volume 2012 (2012), Article ID 846153, 13 pages
http://dx.doi.org/10.1155/2012/846153
Research Article

Modeling and Simulation of MIMO Mobile-to-Mobile Wireless Fading Channels

1Communication Research Laboratory, Faculty of Electrical and Computer Engineering, Yazd University, Yazd 89168-69511, Iran
2Department of Electrical Engineering, University of Isfahan, Isfahan 81746-73441, Iran

Received 24 January 2012; Revised 21 March 2012; Accepted 22 March 2012

Academic Editor: Hon Tat Hui

Copyright © 2012 Gholamreza Bakhshi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Analysis and design of multielement antenna systems in mobile fading channels require a model for the space-time cross-correlation among the links of the underlying multipleinput multiple-output (MIMO) Mobile-to-Mobile (M-to-M) communication channels. In this paper, we propose the modified geometrical two-ring model, a MIMO channel reference model for M-to-M communication systems. This model is based on the extension of single-bounce two-ring scattering model for flat fading channel under the assumption that the transmitter and the receiver are moving. Assuming single-bounce scattering model in both isotropic and nonisotropic environment, a closed-form expression for the space-time cross-correlation function (CCF) between any two subchannels is derived. The proposed model provides an important framework in M-to-M system design, where includes many existing correlation models as special cases. Also, two realizable statistical simulation models are proposed for simulating both isotropic and nonisotropic reference model. The realizable simulation models are based on Sum-of-Sinusoids (SoS) simulation model. Finally, the correctness of the proposed simulation models is shown via different simulation scenarios.

1. Introduction

Mobile-to-Mobile communication channels are expected to play an important role in mobile ad-hoc networks (MANETs), intelligent transportation systems, and relay-based cellular networks, where both the transmitter (Tx) and the receiver (Rx) are in motion. M-to-M channels differ from conventional Base-to-Mobile (B-to-M) cellular radio channels, where the base station (BS) is stationary and relatively free of local scattering.

In a typical macrocell, the BS is elevated and it receives the signal within a narrow beam width, whereas the mobile station (MS) is surrounded by local scatterers. MIMO channel modeling of this typical macrocell environment was investigated in [1, 2]. However, in outdoor microcells, indoor picocells, and M-to-M communication channels, both Tx (BS/MS𝑇) and Rx (MS/MS𝑅) are normally surrounded by local scatterers. Clearly, the MIMO macrocell models of [1, 2] cannot be used for such environments. For these situations, we need a double-directional channel model (see, e.g., [35], in which the double-directional concept is introduced and some measurements results are provided). Akki and Haber [6, 7] showed that the received envelope on M-to-M channels is Rayleigh faded under non line-of-sight (NLoS) condition, but the statistical properties differ from B-to-M channels. They proposed a reference model for single-input single-output (SISO) M-to-M Rayleigh fading channels. Methods for simulating SISO M-to-M channels have been proposed in [8, 9]. Recently, Pätzold et al. have proposed a theoretical reference model for narrow-band MIMO M-to-M communication channels in [1012]. This model is based on geometrical “double-bounce two-ring model” (DBTR) and belongs to the class of double-directional channel models. DBTR model assumes that both Tx and Rx are surrounded by scatterers and each ray is reflected twice. In the other words, in DBTR model, it is assumed that every Tx side scatterer captures the radio signal from Tx and reradiates it in the form of a plane wave to the Rx side scatterers. Then, Rx receives the transmitted radio signal from itself scatterers. A distance-independent DBTR model was proposed in [13] and was simulated in [14]. The main difficulty of DBTR model, discussed in [15, 16], is that the signals reflected by the scatterers at the Rx side are possibly not independent and the channel coefficient may still not be zero mean complex Gaussian. Therefore, the channel covariance matrix cannot completely describe the MIMO channel [16].

This paper proposes a theoretical reference model for MIMO M-to-M Rayleigh fading channels, avoiding difficulty of DBTR model. This model is based on the extension of geometrical “single-bounce two-ring” (SBTR) model proposed in [17] for MIMO B-to-M channel. The SBTR model belongs to the class of double-directional B-to-M channel models. In [17], the authors have avoided many difficulties of the DBTR model. Furthermore the correctness of their model has been shown via real experimental data. Our model, named here modified geometrical two-ring (MGTR), in comparison with SBTR, includes the mobility of both the transmitter and the receiver. Furthermore, we derive a closed-form space-time correlation function for 2D nonisotropic scattering environment. Also, we propose two realizable SoS-based simulation models for simulating the reference model in both isotropic and nonisotropic conditions.

The remainder of the paper is organized as follows. In Section 2, we describe the MGTR, a theoretical reference model for MIMO M-to-M channels. In Section 3, a closed-form expression for the space-time cross-correlation function (STCCF) is derived from our theoretical reference model. Section 4 details the SoS-based simulation models. In Section 5, we describe the calculating methods of simulation model’s parameters. The comparison of the statistical simulation model with the theoretical reference model is presented in Section 6. Finally, concluding remarks are provided in Section 7.

2. Theoretical Reference Model for MIMO M-to-M Channels

In this section, we describe the MGTR model for narrow-band MIMO M-to-M channels. As mentioned before, MGTR is based on the extension of SBTR model in [17], in which both transmitter and receiver are in motion. Consider a narrow-band single-user MIMO communication system with 𝑛𝑇 transmitter and 𝑛𝑅 receiver antenna elements. Assume both Tx and Rx are in motion and equipped with low elevation antennas. The radio propagation environment is characterized by 2D scattering with NLoS conditions between the transmitter and the receiver. The MIMO channel can be described by an 𝑛𝑅×𝑛𝑇 matrix 𝐻(𝑡)=[𝑖𝑗(𝑡)]𝑛𝑅×𝑛𝑇 of complex faded envelopes.

2.1. Geometrical Modified Two-Ring Model

The geometry of modified two-ring model is shown in Figure 1 for a MIMO M-to-M channel with 𝑛𝑇=𝑛𝑅=2 antenna elements, where local scatterers of 𝑀𝑆𝑇 and 𝑀𝑆𝑅 are distributed on two separate rings. The key difference between our model and M-to-M DBTR model is that here only single-bounce rays are considered while in M-to-M DBTR model double-bounce rays have been considered. Therefore, our assumption avoids the problems of DBTR model. Here, we can model the multiple-bounce rays as secondary effects. As can be seen from Figure 1, the local scatterers around the transmitter, denoted by 𝑆𝑘𝑇(𝑘=1,2,,𝑁𝑇), are located on a ring of radius 𝑅, while the local scatterers 𝑆𝑖𝑅(𝑖=1,2,,𝑁𝑅) around the receiver lie on a separate ring of radius 𝑅. The symbols 𝜑𝑇 and 𝜙𝑅 denote the main angle of departure (AOD) and the main angle of arrival (AOA), respectively and the symbols 𝜑𝑅 and 𝜙𝑇 denote the auxiliary AOD and the auxiliary AOA, respectively. It is assumed that the radii 𝑅 and 𝑅 are small in comparison with 𝐷, which is the distance between the transmitter and the receiver (i.e., max{𝑅,𝑅}𝐷). The antenna spacings at the transmitter and the receiver are denoted by 𝛿𝑇 and 𝛿𝑅, respectively. Since the antenna spacing are generally small in comparison with the radii 𝑅 and 𝑅, we might assume that the inequality “min{𝑅,𝑅}max{𝛿𝑇,𝛿𝑅}” is held. The tilt angles between the 𝑥-axis and the orientation of the antenna array at the transmitter and the receiver are denoted by 𝛽𝑇 and 𝛽𝑅, respectively. Moreover, it is assumed that the transmitter and the receiver move with speeds 𝑣𝑇 and 𝑣𝑅 and in direction determined by the angle of motions 𝛼𝑇 and 𝛼𝑅, respectively. Furthermore, 2Δ is the maximum angle spread at 𝑀𝑆𝑇, determined by the scattering around 𝑀𝑆𝑅. Similarly, 2Δ is the maximum angle spread at 𝑀𝑆𝑅, determined by the scattering around 𝑀𝑆𝑇. From Figure 1, it is clear that Δ=arcsin(𝑅/𝐷), and Δ=arcsin(𝑅/𝐷). Note that geometry of our proposed model includes many existing geometrical models.

846153.fig.001
Figure 1: The modified geometrical two-ring model for a 2×2 MIMO channel with scatterers around mobile transmitter 𝑀𝑆𝑇 (left) and mobile receiver 𝑀𝑆𝑅 (right).

However, it must be noted that it is impossible to derive our reference model by fixing one station (Tx or Rx) and inserting the relative velocity of Tx and Rx into other station (Rx or Tx) in the previous B-to-M models. Here, we have two independent clusters of the received paths and, it follows two independent clusters of doppler components due to relative velocity of the Tx and Rx.

2.2. Derivation of the Reference Model

In this subsection, we derive the reference model for the MIMO M-to-M channel. In Figure 1 by considering the forward channel (from 𝑀𝑆𝑇 to 𝑀𝑆𝑅), the 𝑀𝑆𝑅 receives single-bounce rays from both the scatterer 𝑆𝑖𝑅 around the 𝑀𝑆𝑅 and the scatterer 𝑆𝑘𝑇 around the 𝑀𝑆𝑇. For the frequency flat, subchannel between the antenna elements 𝐴𝑝𝑇 and 𝐴𝑙𝑅, 𝑙𝑝(𝑡) denotes the time-varying complex baseband equivalent channel gain. Mathematical representation of the superposition of rays at the 𝐴𝑙𝑅 results in the following expression for the normalized channel gain:𝑙𝑝(𝑡)=𝜂𝑇𝑁𝑇𝑁𝑇𝑘=1exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑘𝑇+𝑑𝑆𝑘𝑇𝐴𝑙𝑅+𝑗Ψ𝑘𝑇+𝑗2𝜋𝑓𝑘1𝑡+𝜂𝑅𝑁𝑅𝑁𝑅𝑖=1exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑖𝑅+𝑑𝑆𝑖𝑅𝐴𝑙𝑅+𝑗Ψ𝑖𝑅+𝑗2𝜋𝑓𝑖2𝑡,(1) where the first and the second summations correspond to the 𝑀𝑆𝑇 and 𝑀𝑆𝑅 rings, respectively. This expression shows the role of AOA and AOD in interrelation between the single-bounce two-ring model in Figure 1 and the 𝑛𝑅×𝑛𝑇 channel transfer matrix 𝐻(𝑡), in which 𝑙𝑝(𝑡) is the element of row 𝑙 and column 𝑝. The 𝑑𝑋𝑌 denotes the distance between 𝑋 and 𝑌, 𝜂𝑇 and 𝜂𝑅 show the respective contributions of scatterers around the 𝑀𝑆𝑇 and 𝑀𝑆𝑅 such that 𝜂𝑇+𝜂𝑅=1. 𝑁𝑇 and 𝑁𝑅 are the number of scatterers around the 𝑀𝑆𝑇 and 𝑀𝑆𝑅, respectively. Ψ𝑘𝑇 and Ψ𝑖𝑅 are the associated phase shifts. Furthermore, as shown in Figure 1, 𝜑𝑘𝑇 and 𝜑𝑖𝑅 are AoD’s of the waves that impinge on 𝑆𝑘𝑇 and 𝑆𝑖𝑅. Similarly 𝜙𝑘𝑇 and 𝜙𝑖𝑅 are AoA’s of the waves scattered from 𝑆𝑘𝑇 and 𝑆𝑖𝑅. Note that 𝑑𝐴𝑝𝑇𝑆𝑘𝑇 and 𝑑𝐴𝑞𝑇𝑆𝑘𝑇 are functions of 𝜑𝑘𝑇, whereas 𝑑𝑆𝑘𝑇𝐴𝑙𝑅 and 𝑑𝑆𝑘𝑇𝐴𝑚𝑅 are functions of 𝜙𝑘𝑇. Other 𝑑𝑋𝑌 can be easily identified from Figure 1. 𝜆 is the wavelength and frequencies 𝑓𝑘1 and 𝑓𝑖2 are given by𝑓𝑘1=𝑓𝑇max𝛼cos𝑇𝜑𝑘𝑇+𝑓𝑅max𝛼cos𝑅𝜙𝑘𝑇,𝑓(2)𝑖2=𝑓𝑇max𝛼cos𝑇𝜑𝑖𝑅+𝑓𝑅max𝛼cos𝑅𝜙𝑖𝑅,(3) where 𝑓𝑇max=𝑣𝑇/𝜆 and 𝑓𝑅max=𝑣𝑅/𝜆 are the maximum Doppler frequencies caused by the movement of the transmitter and the receiver, respectively. We also assume {Ψ𝑘𝑇}𝑁𝑇𝑘=1 and {Ψ𝑖𝑅}𝑁𝑅𝑖=1 are mutually independent and identically distributed (i.i.d) random variables with uniform distributions over [0,2𝜋). According to Figure 1, while 𝜑𝑖𝑅 and 𝜙𝑘𝑇 are dependent to 𝜙𝑖𝑅 and 𝜑𝑘𝑇, respectively, 𝜙𝑖𝑅 and 𝜑𝑘𝑇 are independent variables. In what follows, we call 𝜑𝑘𝑇 the AOD, and 𝜙𝑖𝑅 the AOA.

3. The Space-Time Cross-Correlation Function of the Reference Model

The STCCF plays an important role in MIMO communication channels. In this section, we derive a closed-form expression for STCCF. The normalized STCC between two subchannel gains 𝑙𝑝(𝑡) and 𝑚𝑞(𝑡) is defined by 𝜌𝑙𝑝,𝑚𝑞(𝜏)=𝔼[𝑙𝑝(𝑡)𝑚𝑞(𝑡+𝜏)], where 𝔼() is the statistical expectation operator and () denotes complex conjugate operation. Based on independent properties of Ψ𝑘𝑇 and Ψ𝑖𝑅, it can be asymptotically written by𝜌𝑙𝑝,𝑚𝑞(𝜏)=lim𝑁𝑇𝜂𝑇𝑁𝑇𝑁𝑇𝑘=1𝔼×exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑘𝑇𝑑𝐴𝑞𝑇𝑆𝑘𝑇+𝑑𝑆𝑘𝑇𝐴𝑙𝑅𝑑𝑆𝑘𝑇𝐴𝑚𝑅𝑗2𝜋𝑓𝑘1𝜏+lim𝑁𝑅𝜂𝑅𝑁𝑅𝑁𝑅𝑖=1𝔼×exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑖𝑅𝑑𝐴𝑞𝑇𝑆𝑖𝑅+𝑑𝑆𝑖𝑅𝐴𝑙𝑅𝑑𝑆𝑖𝑅𝐴𝑚𝑅𝑗2𝜋𝑓𝑖2𝜏.(4) For large 𝑁𝑇 and 𝑁𝑅, the discrete AoDs, 𝜑𝑘𝑇, and the discrete AoAs, 𝜙𝑖𝑅, can be replaced with their continuous random variables 𝜑𝑇 and 𝜙𝑅 with probability density functions (pdf) 𝑓𝑀𝑆𝑇(𝜑𝑇) and 𝑓𝑀𝑆𝑅(𝜙𝑅), respectively. Therefore, (4) can be reduced to the following integral form:𝜌𝑙𝑝,𝑚𝑞(𝜏)=𝜂𝑇𝜋𝜋exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑇𝑑𝐴𝑞𝑇𝑆𝑇+𝑑𝑆𝑇𝐴𝑙𝑅𝑑𝑆𝑇𝐴𝑚𝑅𝑗2𝜋𝑓1𝜏𝑓𝑀𝑆𝑇𝜑𝑇𝑑𝜑𝑇+𝜂𝑅𝜋𝜋exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑅𝑑𝐴𝑞𝑇𝑆𝑅+𝑑𝑆𝑅𝐴𝑙𝑅𝑑𝑆𝑅𝐴𝑚𝑅𝑗2𝜋𝑓2𝜏𝑓𝑀𝑆𝑅𝜙𝑅𝑑𝜙𝑅,(5) where 𝑓1 and 𝑓2 are the continuous form of 𝑓𝑘1 and 𝑓𝑖2 in (2) and (3), respectively. All of the 𝑑𝑋𝑌’s in first integral of (5) depend on 𝜑𝑇 and in the second integral depend on 𝜙𝑅.

Based on the application of the law of cosines in appropriate triangles in Figure 1, and assumption min{𝑅,𝑅}max{𝛿𝑅,𝛿𝑇}, we obtain the following approximation:𝑑𝐴𝑝𝑇𝑆𝑇𝑑𝐴𝑞𝑇𝑆𝑇𝛿𝑇𝑝𝑞𝛽cos𝑇𝜑𝑇,𝑑𝑆𝑇𝐴𝑙𝑅𝑑𝑆𝑇𝐴𝑚𝑅𝛿𝑅𝑙𝑚𝛽cos𝑅𝜙𝑇,𝑑𝐴𝑝𝑇𝑆𝑅𝑑𝐴𝑞𝑇𝑆𝑅𝛿𝑇𝑝𝑞𝛽cos𝑇𝜑𝑅,𝑑𝑆𝑅𝐴𝑙𝑅𝑑𝑆𝑅𝐴𝑚𝑅𝛿𝑅𝑙𝑚𝛽cos𝑅𝜙𝑅.(6) Now we apply the law of sines and obtain the following identities:𝐷𝜙sin𝑇𝜑𝑇=𝑅sin𝜋𝜙𝑇,𝐷𝜙sin𝑅𝜑𝑅=𝑅𝜑sin𝑅.(7) Based on the assumption max{𝑅,𝑅}𝐷, we conclude that Δ𝑅/𝐷, and Δ𝑅/𝐷. This observation, together with sin𝜖𝜖 when 𝜖 is small, considering 𝜙𝑇 is almost 𝜋 and 𝜑𝑅 is almost 0, allows us to derive the following approximations from (7):𝜙𝑇𝜋Δsin𝜑𝑇,𝜑𝑅Δsin𝜙𝑅.(8) Furthermore, using sin𝜖𝜖 and cos𝜖1 when 𝜖 is small, together with (8), the following approximations are derived:𝛽cos𝑅𝜙𝑇cos𝛽𝑅+Δsin𝛽𝑅sin𝜑𝑇𝛼,(9)cos𝑅𝜙𝑇cos𝛼𝑅+Δsin𝛼𝑅sin𝜑𝑇𝛽,(10)cos𝑇𝜑𝑅cos𝛽𝑇+Δsin𝛽𝑇sin𝜙𝑅𝛼,(11)cos𝑇𝜑𝑅cos𝛼𝑇+Δsin𝛼𝑇sin𝜙𝑅.(12)

Now, by substituting (10) and (12) to continuous form of (2) and (3), respectively, the following approximations are derived:𝑓1𝑓𝑇max𝛼cos𝑇𝜑𝑇𝑓𝑅maxcos𝛼𝑅+𝑓𝑅maxΔsin𝜑𝑇sin𝛼𝑅,𝑓2𝑓𝑇maxcos𝛼𝑇+𝑓𝑇maxΔsin𝜙𝑅sin𝛼𝑇+𝑓𝑅max𝛼cos𝑅𝜙𝑅.(13)

For any given 𝑓𝑀𝑆𝑇() and 𝑓𝑀𝑆𝑅(), the right-hand side (RHS) of (5) can be calculated numerically, using the trigonometric function relationships given in (6). Note that the RHS of (5) includes two parts. The first part corresponds to STCC contributed by the scattering ring around the 𝑀𝑆𝑇, and the second part comes from the scattering ring around the 𝑀𝑆𝑅. Given the assumptions max{𝑅,𝑅}𝐷 and min{𝑅,𝑅}max{𝛿𝑅,𝛿𝑇}, by plugging (6), (9) and (11) into (5), equation (5) is approximated by𝜌𝑙𝑝,𝑚𝑞(𝜏)𝜂𝑇𝜋𝜋𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞𝛽cos𝑇𝜑𝑇+𝛿𝑅𝑙𝑚×cos𝛽𝑅+Δsin𝛽𝑅sin𝜑𝑇𝑗2𝜋𝑓1𝜏𝑓𝑀𝑆𝑇𝜑𝑇𝑑𝜑𝑇+𝜂𝑅𝜋𝜋𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞cos𝛽𝑇+Δsin𝛽𝑇sin𝜙𝑅+𝛿𝑅𝑙𝑚𝛽cos𝑅𝜙𝑅𝑗2𝜋𝑓2𝜏𝑓𝑀𝑆𝑅𝜙𝑅𝑑𝜙𝑅.(14)

Now, we consider the nonisotropic scattering. Prior works use several different scatterer distributions, included uniform, Gaussian, Laplacian, and von Mises. In this section, we use the von Mises distribution because the measurement experiences show that it approximates many of the previously mentioned distributions. The von Mises pdf is defined by [18]:1𝑝(𝜃)=2𝜋𝐼0[](𝑘)exp𝑘cos(𝜃𝜇),(15) where 𝐼0() is the zeroth-order modified Bessel function of the first kind, 𝜇[𝜋,𝜋) is the mean angle of scatterers’ distribution on the ring, and 𝑘 controls the spread of scatterers around the mean. When 𝑘=0, 𝑝(𝜃)=1/(2𝜋) is a uniform distribution yielding 2D isotropic scattering. As 𝑘 increases, the scatterers become more clustered around angle 𝜇 and the scattering becomes increasingly nonisotropic. Therefor, the von Mises pdf of AOD and AOA is given by 𝑓𝑀𝑆𝑇(𝜑𝑇)=exp[𝑘𝑇cos(𝜑𝑇𝜇𝑇)]/(2𝜋𝐼0(𝑘𝑇)) and 𝑓𝑀𝑆𝑅(𝜙𝑅)=exp[𝑘𝑅cos(𝜙𝑅𝜇𝑅)]/(2𝜋𝐼0(𝑘𝑅)), respectively.

From [[19], eq. 3.338], we have𝜋𝜋exp(𝑥sin𝜃+𝑦cos𝜃)𝑑𝜃=2𝜋𝐼0𝑥2+𝑦2.(16) Under nonisotropic conditions, and by substituting (13) into (14) and calculating the two integrals of (14) by (16), the STCCF of our reference model is derived after some algebraic manipulations (see (17)). 𝜌𝑙𝑝,𝑚𝑞𝜂(𝜏)𝑇𝐼0𝑘𝑇exp𝑗2𝜋𝜆𝛿𝑅𝑙𝑚cos𝛽𝑅+𝑗2𝜋𝑓𝑅max𝜏cos𝛼𝑅×𝐼0𝑘2𝑇2𝜋𝜆2𝛿𝑅𝑙𝑚Δsin𝛽𝑅𝛿𝑅𝑙𝑚Δsin𝛽𝑅+2𝛿𝑇𝑝𝑞sin𝛽𝑇(2𝜋)2𝜆𝛿𝑇𝑝𝑞𝛿𝑇𝑝𝑞𝜆2𝑓𝑇max𝛼𝜏cos𝑇𝛽𝑇2𝜋𝑓𝑇max𝜏2+2(2𝜋)2𝜆𝛿𝑅𝑙𝑚Δ𝜏sin𝛽𝑅𝑓𝑇maxsin𝛼𝑇+Δ𝑓𝑅maxsin𝛼𝑅(2𝜋)2×𝑓𝑅max𝜏Δsin𝛼𝑅𝑓𝑅max𝜏Δsin𝛼𝑅2𝜆𝛿𝑇𝑝𝑞sin𝛽𝑇+2𝑓𝑇max𝜏sin𝛼𝑇+𝑗2𝑘𝑇2𝜋𝜆𝛿𝑇𝑝𝑞𝛽cos𝑇𝜇𝑇2𝜋𝑓𝑇max𝛼𝜏cos𝑇𝜇𝑇+2𝜋𝜆𝛿𝑅𝑙𝑚Δsin𝛽𝑅sin𝜇𝑇2𝜋𝑓𝑅max𝜏Δsin𝛼𝑅sin𝜇𝑇1/2+𝜂𝑅𝐼0𝑘𝑅𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞cos𝛽𝑇𝑗2𝜋𝑓𝑇max𝜏cos𝛼𝑇×𝐼0𝑘2𝑅2𝜋𝜆2𝛿𝑇𝑝𝑞Δsin𝛽𝑇𝛿𝑇𝑝𝑞Δsin𝛽𝑇+2𝛿𝑅𝑙𝑚sin𝛽𝑅(2𝜋)2𝜆𝛿𝑅𝑙𝑚𝛿𝑅𝑙𝑚𝜆2𝑓𝑅max𝛼𝜏cos𝑅𝛽𝑅2𝜋𝑓𝑅max𝜏2+2(2𝜋)2𝜆𝛿𝑇𝑝𝑞Δ𝜏×sin𝛽𝑇𝑓𝑅maxsin𝛼𝑅+Δ𝑓𝑇maxsin𝛼𝑇(2𝜋)2𝑓𝑇max𝜏Δsin𝛼𝑇𝑓𝑇max𝜏Δsin𝛼𝑇2𝜆𝛿𝑅𝑙𝑚sin𝛽𝑅+2𝑓𝑅max𝜏sin𝛼𝑅+𝑗2𝑘𝑅2𝜋𝜆𝛿𝑅𝑙𝑚𝛽cos𝑅𝜇𝑅2𝜋𝑓𝑅max𝛼𝜏cos𝑅𝜇𝑅+2𝜋𝜆𝛿𝑇𝑝𝑞Δsin𝛽𝑇sin𝜇𝑅2𝜋𝑓𝑇max𝜏Δsin𝛼𝑇sin𝜇𝑅1/2.(17)

Note that many existing correlation functions are special cases of our MIMO M-to-M space-time correlation function in (17). For example: (i)For 2-D isotropic scattering around both 𝑀𝑆𝑇 and 𝑀𝑆𝑅 (𝑘𝑇=𝑘𝑅=0), the STCCF of our reference model reduces to STCCF of MGTR model in isotropic environment [20] as𝜌𝑙𝑝,𝑚𝑞(𝜏)𝜂𝑇exp𝑗2𝜋𝜆𝛿𝑅𝑙𝑚cos𝛽𝑅+𝑗2𝜋𝑓𝑅max𝜏cos𝛼𝑅×𝐼02𝜋𝜆2𝛿𝑅𝑙𝑚Δsin𝛽𝑅𝛿𝑅𝑙𝑚Δsin𝛽𝑅+2𝛿𝑇𝑝𝑞sin𝛽𝑇(2𝜋)2𝜆𝛿𝑇𝑝𝑞𝛿𝑇𝑝𝑞𝜆2𝑓𝑇max𝛼𝜏cos𝑇𝛽𝑇2𝜋𝑓𝑇max𝜏2+2(2𝜋)2𝜆𝛿𝑅𝑙𝑚Δ𝜏sin𝛽𝑅𝑓𝑇maxsin𝛼𝑇+Δ𝑓𝑅maxsin𝛼𝑅(2𝜋)2𝑓𝑅max𝜏Δsin𝛼𝑅𝑓𝑅max𝜏Δsin𝛼𝑅2𝜆𝛿𝑇𝑝𝑞sin𝛽𝑇+2𝑓𝑇max𝜏sin𝛼𝑇1/2+𝜂𝑅𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞cos𝛽𝑇𝑗2𝜋𝑓𝑇max𝜏cos𝛼𝑇×𝐼02𝜋𝜆2𝛿𝑇𝑝𝑞Δsin𝛽𝑇𝛿𝑇𝑝𝑞Δsin𝛽𝑇+2𝛿𝑅𝑙𝑚sin𝛽𝑅(2𝜋)2𝜆𝛿𝑅𝑙𝑚𝛿𝑅𝑙𝑚𝜆2𝑓𝑅max𝛼𝜏cos𝑅𝛽𝑅2𝜋𝑓𝑅max𝜏2+2(2𝜋)2𝜆𝛿𝑇𝑝𝑞Δ𝜏sin𝛽𝑇𝑓𝑅maxsin𝛼𝑅+Δ𝑓𝑇maxsin𝛼𝑇(2𝜋)2𝑓𝑇max𝜏Δsin𝛼𝑇𝑓𝑇max𝜏Δsin𝛼𝑇2𝜆𝛿𝑅𝑙𝑚sin𝛽𝑅+2𝑓𝑅max𝜏sin𝛼𝑅1/2.(18)(ii)For stationary 𝑀𝑆𝑇 (𝑓𝑇max=0), the STCCF of our reference model reduces to MIMO B-to-M communication channel model based on single-bounce two-ring model proposed in [17, equation (7)].(iii)If there is no scattering around the 𝑀𝑆𝑇 such in a macrocell (𝜂𝑇=0) and stationary 𝑀𝑆𝑇 (𝑓𝑇max=0), (17) is simplified to STCCF of the conventional “one-ring” model for MIMO B-to-M communication channel proposed in [2]. In these conditions, the first half of (17) disappears, and the remaining part is the same as (12) in [2].(iv)If there is no scattering around the 𝑀𝑆𝑇 such in a macrocell (𝜂𝑇=0), stationary 𝑀𝑆𝑇 (𝑓𝑇max=0) and, with 𝑙=𝑚 and 𝑝=𝑞, our reference model’s STCCF is simplified to conventional “one-ring” model for SISO B-to-M communication channel. This reduces (17) to the well-known Clarke’s temporal correlation function, that is, 𝐽0(2𝜋𝑓𝑅max𝜏) [21], where 𝐽0() is the Bessel function of the first kind of zero order.

4. The Simulation Model

In this section, we derive a statistical simulation model. The theoretical model proposed in Section 2 assumes an infinite number of scatterers, which prevents practical implementation. Actually, in a practical communication channels, the number of scatterers is finite. In the following, we propose a SoS-based statistical simulation model that matches the statistical properties of the theoretical reference model.

Generally, SoS models [22] approximate the underlying random processes by the superposition of a finite number of properly selected functions and can be classified as either statistical or deterministic. In other words, the SoS models are based on a superposition of an infinite number of weighted harmonic functions with equidistant frequencies and random phases. Actually, the SoS models are applied by using only a finite number of harmonic functions for simulating the communication channels [22]. Deterministic SoS models have sinusoids with fixed phases, amplitudes, and Doppler frequencies for all simulation trials. Statistical SoS models leave at least one of the parameter sets (amplitudes, phases, or Doppler frequencies) as random variables that vary with each simulation trial.

The following function is considered as the complex-faded envelope in a real environment that contains finite number of scatterers (finite number of harmonic functions):𝑙𝑝(𝑡)=𝜂𝑇𝑁𝑇𝑁𝑇𝑘=1exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑘𝑇+𝑑𝑆𝑘𝑇𝐴𝑙𝑅+𝑗Ψ𝑘𝑇+𝑗2𝜋𝑓𝑘1𝑡+𝜂𝑅𝑁𝑅𝑁𝑅𝑖=1exp𝑗2𝜋𝜆𝑑𝐴𝑝𝑇𝑆𝑖𝑅+𝑑𝑆𝑖𝑅𝐴𝑙𝑅+𝑗Ψ𝑖𝑅+𝑗2𝜋𝑓𝑖2𝑡,(19) where the parameters of above equation are defined in Section 2.2. In contrast to the reference model, the discrete AODs 𝜑𝑘𝑇 and AOAs 𝜙𝑖𝑅 are now constant, which will be determined in Section 5. The phases Ψ𝑘𝑇 and Ψ𝑖𝑅 are still i.i.d. random variables, each with uniform distribution on the interval [0,2𝜋). Hence, 𝑙𝑝(𝑡) represents a stochastic process. The STCCF between 𝑙𝑝(𝑡) and 𝑚𝑞(𝑡) is defined as ̂𝜌𝑙𝑝,𝑚𝑞(𝜏)=𝔼[𝑙𝑝(𝑡)𝑚𝑞(𝑡+𝜏)], where () denotes the complex conjugate operation, and 𝔼() is the statistical expectation operator, which applies to the random phases Ψ𝑘𝑇 and Ψ𝑖𝑅. It can be shown that STCCF can be expressed in closed form, considering finite scatterers around the 𝑀𝑆𝑇 and 𝑀𝑆𝑅 (finite number of harmonic functions), aŝ𝜌𝑙𝑝,𝑚𝑞𝜂(𝜏)𝑇𝑁𝑇𝑁𝑇𝑘=1×𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞𝛽cos𝑇𝜑𝑘𝑇+𝛿𝑅𝑙𝑚×cos𝛽𝑅+Δsin𝛽𝑅sin𝜑𝑘𝑇𝑗2𝜋𝑓𝑘1𝜏+𝜂𝑅𝑁𝑅𝑁𝑅𝑖=1𝑗exp2𝜋𝜆𝛿𝑇𝑝𝑞cos𝛽𝑇+Δsin𝛽𝑇sin𝜙𝑖𝑅+𝛿𝑅𝑙𝑚𝛽cos𝑅𝜙𝑖𝑅𝑗2𝜋𝑓𝑖2𝜏,(20) where𝑓𝑘1𝑓𝑇max𝛼cos𝑇𝜑𝑘𝑇𝑓𝑅maxcos𝛼𝑅+𝑓𝑅maxΔsin𝜑𝑘𝑇sin𝛼𝑅𝑓𝑖2𝑓𝑇maxcos𝛼𝑇+𝑓𝑇maxΔsin𝜙𝑖𝑅sin𝛼𝑇+𝑓𝑅max𝛼cos𝑅𝜙𝑖𝑅.(21)

In the following section we introduce two methods for determining the constant discrete AODs 𝜑𝑘𝑇 and AOAs 𝜙𝑖𝑅.

5. Parameters Calculation of Simulation Model

In this section, we present two methods for the computation of the parameters determining the statistics of the MIMO channel simulation model. The first method is the method of exact Doppler spread (MEDS), which is recommended in case of isotropic scattering. The second method is the 𝐿𝑝-Norm method. This method can be applied for any given distribution of the local scatterers, such as the Gaussian distribution, the Laplacian distribution, and the von Mises distribution. In other words, the 𝐿𝑝-Norm method is a general method for calculation of the parameters of deterministic simulation models.

5.1. Method of Exact Doppler Spread (MEDS)

The MEDS method was first time proposed in [23], which is recommended in case of isotropic scattering, and was also described in [22] in details. This method is extended in [11, 12, 14] for simulating the MIMO M-to-M DBTR reference model. According to MEDS method the discrete AODs 𝜑𝑘𝑇 and AOAs 𝜙𝑖𝑅 are determined by [22]:𝜑𝑘𝑇=2𝜋𝑁𝑇1𝑘2,𝑘=1,2,,𝑁𝑇,𝜙𝑖𝑅=2𝜋𝑁𝑅1𝑖2,𝑖=1,2,,𝑁𝑅.(22) Therefore, in the statistical simulation model only phases Ψ𝑘𝑇 and Ψ𝑖𝑅 are random parameters. They are i.i.d. random variables uniformly distributed over [0,2𝜋).

5.2. 𝐿𝑝-Norm Method

When the AODs 𝜑𝑘𝑇 and AOAs 𝜙𝑖𝑅 are nonuniformly distributed on rings around the transmitter and the receiver, respectively, the recommend method for determining the AODs and AOAs is 𝐿𝑝-Norm which is described in detail in [22]. This method is extended in [11, 12, 14] for simulating the MIMO M-to-M DBTR reference model in nonisotropic environment. According to 𝐿𝑝-Norm method, the discrete AODs 𝜑𝑘𝑇 and AOAs 𝜙𝑖𝑅 are determined by minimizing the following error norm [22]:𝐸𝜌(𝑝)𝑙𝑝,𝑚𝑞=1𝜏max𝜏max0||𝜌𝑙𝑝,𝑚𝑞(𝜏)̂𝜌𝑙𝑝,𝑚𝑞||(𝜏)𝑝𝑑𝜏1/𝑝,(23) where 𝑝=1,2,, 𝜌𝑙𝑝,𝑚𝑞(𝜏) and ̂𝜌𝑙𝑝,𝑚𝑞(𝜏) are the STCCF of the theoretical reference model in nonisotropic condition (17) and the 𝐿𝑝-Norm simulation model, respectively. Therefore, in the statistical simulation model only phases Ψ𝑘𝑇 and Ψ𝑖𝑅 are random parameters. They are i.i.d random variables uniformly distributed over [0,2𝜋).

6. Performance Evaluation of the Simulation Models

This section evaluates the performance of the simulation models by comparing its statistical properties with those of the theoretical model. In all simulations, The following parameters were chosen for both models. The antenna tilt angles 𝛽𝑇 and 𝛽𝑅 were defined as 𝛽𝑇=𝛽𝑅=𝜋/2. At the transmitter side, the angle of motion 𝛼𝑇 was set to 𝜋/4, while the receiver was moving at an angle of 𝛼𝑅=0. Identical maximum Doppler frequencies 𝑓𝑇max=𝑓𝑅max=91Hz were assumed, and the wavelength 𝜆 was set to 𝜆=0.15m (according to [10, 11]). Furthermore, the other parameters have their quantity as Δ=𝜋/3, Δ=𝜋/6, 𝜇𝑇=5𝜋/8, 𝜇𝑅=0, and 𝜂𝑅=0.2, according to Table I of [17].

6.1. The MEDS Simulation Model

Such as mentioned before, the model parameters 𝜑𝑘𝑇 and 𝜙𝑖𝑅 have been determined by the MEDS method, since we assume isotropic scattering around both the transmitter and the receiver (𝑘𝑇=𝑘𝑅=0). Note that the mismatch criteria in the following simulation is Relative Error and is set to 103. The relative error is defined by𝜖𝜌𝑙𝑝,𝑚𝑞=1/𝜏max𝜏max0||𝜌𝑙𝑝,𝑚𝑞(𝜏)̂𝜌𝑙𝑝,𝑚𝑞||(𝜏)2𝑑𝜏1/21/𝜏max𝜏max0||𝜌𝑙𝑝,𝑚𝑞||(𝜏)2𝑑𝜏1/2,(24) where 𝜌𝑙𝑝,𝑚𝑞(𝜏) and ̂𝜌𝑙𝑝,𝑚𝑞(𝜏) are the STCCF of the theoretical reference model and the simulation model, respectively. Now we consider two simulation scenario as follows.(i)First Scenario. In this scenario, we compare the temporal autocorrelation function (ACF) of the MEDS simulation model with the temporal ACF of theoretical isotropic reference model, derived in Section 3 (18). Figure 2 shows this comparison for {𝑁𝑇=𝑁𝑅=20,30,40,50}. This figure shows that the temporal ACF of the MEDS simulation model is matched to the temporal ACF of theoretical isotropic reference model until a limited normalized time delay, that is shown in the subfigures and we call it the Matched Time. In other words, the matched time is the maximum normalized time delay that until it relative error between the reference and the MEDS simulation model is negligible. The matched time is dependent on the number of scatterers and the relative error. It is denoted by [𝑓max𝜏]max, where 𝑓max=𝑓𝑇max=𝑓𝑅max. As evident from the simulations, the maximum time delay 𝜏max is a key parameter for the proposed MEDS simulation model and requires to be set properly to use it for simulating the isotropic MGTR reference model. Also, by increasing the number of scatterers (the number of harmonic functions), 𝑁𝑇 and 𝑁𝑅, the matched time increases.(ii)Second Scenario. In this scenario, we compare the STCCF of the simulation model (̂𝜌𝑙𝑝,𝑚𝑞(𝜏)) with the STCCF of the theoretical isotropic reference model (𝜌𝑙𝑝,𝑚𝑞(𝜏)) for 𝛿𝑇𝑝𝑞=𝛿𝑅𝑙𝑚=1𝜆. Figure 3 denotes this comparison for {𝑁𝑇=𝑁𝑅=20,30,40,50}. It is evident, like the first scenario, by increasing the number of scatterers (the number of harmonic functions), 𝑁𝑇 and 𝑁𝑅, the matched time increases.

fig2
Figure 2: The normalized temporal ACF of the isotropic reference model and the MEDS simulation model for 𝑁𝑇=𝑁𝑅=20,30,40,50.
fig3
Figure 3: The STCCF of the isotropic reference model and the MEDS simulation model for 𝛿𝑇𝑝𝑞=𝛿𝑅𝑙𝑚=1𝜆 and 𝑁𝑇=𝑁𝑅=20,30,40,50.
6.2. The 𝐿𝑝-Norm Simulation Model

For performance evaluation of 𝐿𝑝-Norm simulation model, first we determine the optimum parameters 𝜑𝑘𝑇 and 𝜙𝑖𝑅 by minimizing the error norm defined in (23). Note that in the following simulation scenarios, we assume 𝑘𝑇=0.5 and 𝑘𝑅=0, according to Table I of [17]. Also, we assume 𝑝=2, actually, we minimize the 𝐿2-Norm. Now, we consider two scenarios like the MEDS simulation model scenarios. First, we compare the temporal ACF of the 𝐿𝑝-Norm simulation model with the temporal ACF of theoretical nonisotropic reference model, derived in Section 3 (17) for {𝑁𝑇=𝑁𝑅=20,30,40,50}. Figures 4 and 5 show this comparison for real part and imaginary part of temporal ACF, respectively. Second, we compare the STCCF of the 𝐿𝑝-Norm simulation model with the STCCF of theoretical nonisotropic reference model, for 𝛿𝑇𝑝𝑞=𝛿𝑅𝑙𝑚=1𝜆 and {𝑁𝑇=𝑁𝑅=20,30,40,50}. This comparison result is shown in Figures 6 and 7 for real part and imaginary part of STCCF, respectively.

fig4
Figure 4: The normalized temporal ACF of the nonisotropic reference model and the 𝐿𝑝-Norm simulation model for 𝑁𝑇=𝑁𝑅=20,30,40,50 (real part).
fig5
Figure 5: The normalized temporal ACF of the nonisotropic reference model and the 𝐿𝑝-Norm simulation model for 𝑁𝑇=𝑁𝑅=20,30,40,50 (imaginary part).
fig6
Figure 6: The STCCF of the nonisotropic reference model and the 𝐿𝑝-Norm simulation model for 𝛿𝑇𝑝𝑞=𝛿𝑅𝑙𝑚=1𝜆 and 𝑁𝑇=𝑁𝑅=20,30,40,50 (real part).
fig7
Figure 7: The STCCF of the nonisotropic reference model and the 𝐿𝑝-Norm simulation model for 𝛿𝑇𝑝𝑞=𝛿𝑅𝑙𝑚=1𝜆 and 𝑁𝑇=𝑁𝑅=20,30,40,50 (imaginary part).

Note that the number displayed on the Figures 47 is the minimized relative error of 𝐿𝑝-Norm method and defined as𝐸relativeerror=𝜌(2)𝑙𝑝,𝑚𝑞1/𝜏max𝜏max0||𝜌𝑙𝑝,𝑚𝑞||(𝜏)2𝑑𝜏1/2.(25)

It must be mentioned, unlike the MEDS method, the 𝐿𝑝-Norm method has an advantage that can be applied to any given distribution of the local scatterers (it is useful for determining the nonuniform distributed discrete parameters AODs and AOAs). But, it has more complexity in comparison with the MEDS method. In MEDS method, the Matched Time is depended on the relative error and the number of weighted harmonic functions (the number of scatterers around the transmitter and the receiver, 𝑁𝑇 and 𝑁𝑅). In the 𝐿𝑝-Norm method, the minimization is performed over interval [0,𝜏max] and the maximum Matched Time is equal to 𝜏max for predefined constants 𝑁𝑇 and 𝑁𝑅 that by increasing them the minimization error is decreased.

7. Conclusion

This paper proposed a theoretical reference model for Rayleigh fading MIMO M-to-M channels. This reference model was based on the extension of single-bounce two-ring model that avoids the technical difficulties of the double-bounce two-ring model. The closed-form cross-correlation function for 2D nonisotropic scattering was derived for this proposed reference model. The presented model is an extension of M-to-M channel model proposed by Akki and Haber with respect to multiple antenna at the transmitter and the receiver. Moreover, it includes the single-bounce two-ring MIMO channel model introduced by Wang et al. as a special case when the transmitter is fixed and only the receiver is moving. Also, we propose two efficient and realizable statistical simulation models for simulating the theoretical reference model in both isotropic and nonisotropic conditions. The correctness of proposed simulation models was shown via different simulation trials.

Acknowledgment

The financial support from the Iran Telecommunication Research Center (ITRC) is gratefully acknowledged.

References

  1. D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading correlation and its effect on the capacity of multielement antenna systems,” IEEE Transactions on Communications, vol. 48, no. 3, pp. 502–513, 2000. View at Publisher · View at Google Scholar · View at Scopus
  2. A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE Journal on Selected Areas in Communications, vol. 20, no. 3, pp. 550–560, 2002. View at Publisher · View at Google Scholar · View at Scopus
  3. M. Steinbauer, D. Hampicke, G. Sommerkorn, et al., “Array measurement of the doubledirectional mobile radio channel,” in Proceedings of the IEEE Vehicular Technology Conference, pp. 1656–1662, Tokyo, Japan, 2000.
  4. M. Steinbauer, A. F. Molisch, and E. Bonek, “The double-directional radio channel,” IEEE Antennas and Propagation Magazine, vol. 43, no. 4, pp. 51–63, 2001. View at Publisher · View at Google Scholar · View at Scopus
  5. D. Gesbert, H. Bölcskei, D. A. Gore, and A. J. Paulraj, “Outdoor MIMO wireless channels: models and performance prediction,” IEEE Transactions on Communications, vol. 50, no. 12, pp. 1926–1934, 2002. View at Publisher · View at Google Scholar · View at Scopus
  6. A. S. Akki and F. Haber, “A statistical model of radio mobile-to-mobile land communication channel,” IEEE Transactions on Vehicular Technology, vol. 35, no. 1, pp. 2–7, 1986. View at Scopus
  7. A. S. Akki, “Statistical properties of mobile-to-mobile land communication channels,” IEEE Transactions on Vehicular Technology, vol. 43, no. 4, pp. 826–831, 1994. View at Publisher · View at Google Scholar · View at Scopus
  8. C. S. Patel, G. L. Stüber, and T. G. Pratt, “Simulation of Rayleigh-faded mobile-to-mobile communication channels,” IEEE Transactions on Communications, vol. 53, no. 11, pp. 1876–1884, 2005. View at Publisher · View at Google Scholar · View at Scopus
  9. A. G. Zajić and G. L. Stüber, “A new simulation model for mobile-to-mobile rayleigh fading channels,” in Proceedings of the IEEE Wireless Communications and Networking Conference (WCNC '06), pp. 1266–1270, Las Vegas, Nev, USA, April 2006. View at Scopus
  10. M. Pätzold, B. O. Hogstad, N. Youssef, and D. Kim, “A MIMO mobileto-mobile channel model: part I-the reference model,” in Proceedings of the Personal, Indoor and Mobile Radio Communications (PIMRC '05), pp. 573–578, Berlin, Germany, September 2005.
  11. B. O. Hogstad, M. Pätzold, N. Youssef, and D. Kim, “A MIMO mobileto-mobile channel model: part II-the simulation model,” in Proceedings of the Personal, Indoor and Mobile Radio Communications (PIMRC '05), pp. 562–567, Berlin, Germany, September 2005.
  12. M. Pätzold, B. O. Hogstad, and N. Youssef, “Modeling, analysis, and simulation of MIMO mobile-to-mobile fading channels,” IEEE Transactions on Wireless Communications, vol. 7, no. 2, pp. 510–520, 2008. View at Publisher · View at Google Scholar · View at Scopus
  13. A. G. Zajić and G. L. Stüber, “Space-time correlated MIMO mobile-to-mobile channels,” in Proceedings of the IEEE 17th International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC '06), Helsinki, Finland, September 2006. View at Publisher · View at Google Scholar · View at Scopus
  14. A. G. Zajić and G. L. Stüber, “Simulation models for MIMO mobileto-mobile channels,” in Proceedings of the IEEE Military Communications Conference (MILCOM '06), pp. 1–7, Washington, DC, USA, October 2006.
  15. K. Yu and B. Ottersten, “Models for MIMO propagation channels: a review,” Wireless Communications and Mobile Computing, vol. 2, no. 7, pp. 653–666, 2002. View at Publisher · View at Google Scholar · View at Scopus
  16. K. Yu, Multiple-input multiple-output radio propagation channels: characteristics and models, Doctoral thesis, Signals, Sensors and Systems, Royal Institute of Technology (KTH), 2005.
  17. S. Wang, A. Abdi, J. Salo et al., “Time-varying MIMO channels: parametric statistical modeling and experimental results,” IEEE Transactions on Vehicular Technology, vol. 56, no. 4, pp. 1949–1963, 2007. View at Publisher · View at Google Scholar · View at Scopus
  18. A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station,” IEEE Transactions on Vehicular Technology, vol. 51, no. 3, pp. 425–434, 2002. View at Publisher · View at Google Scholar · View at Scopus
  19. I. S. Gradshteyn and I. M. Ryzhik, “Table of Integral, Series and Products,” A. Jeffrey, Ed., Academic Press, San Diego, Calif, USA, 5th edition, 1994.
  20. G. Bakhshi, R. Saadat, and K. Shahtalebi, “A modified two-ring reference model for MIMO mobile-to-mobile communication channels,” in Proceedings of the International Symposium on Telecommunications (IST '08), pp. 409–413, Tehran, Iran, August 2008. View at Publisher · View at Google Scholar · View at Scopus
  21. R. H. Clarke, “A statistical theory of mobile-radio reception,” Bell System Technical Journal, vol. 47, pp. 957–1000, 1968.
  22. M. Pätzold, Mobile Fading Channels, John Wiley & Sons, Chichester, UK, 2002.
  23. M. Pätzold, U. Killat, F. Laue, and Y. Li, “On the statistical properties of deterministic simulation models for mobile fading channels,” IEEE Transactions on Vehicular Technology, vol. 47, no. 1, pp. 254–269, 1998. View at Scopus