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., [3–5], 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, PΓ€tzold et al. have proposed a theoretical reference model for narrow-band MIMO M-to-M communication channels in [10–12]. This model is based on geometrical β€œdouble-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 β€œsingle-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(𝑑)=[β„Žπ‘–π‘—(𝑑)]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 π‘†π‘˜π‘‡(π‘˜=1,2,…,𝑁𝑇), are located on a ring of radius π‘…ξ…ž, while the local scatterers 𝑆𝑖𝑅(𝑖=1,2,…,𝑁𝑅) around the receiver lie on a separate ring of radius 𝑅. The symbols πœ‘π‘‡ and πœ™π‘… denote the main angle of departure (AOD) and the main angle of arrival (AOA), respectively and the symbols πœ‘π‘… and πœ™π‘‡ 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′}β‰ͺ𝐷). 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 β€œmin{𝑅,𝑅′}≫max{𝛿𝑇,𝛿𝑅}” is held. The tilt angles between the π‘₯-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, 2Ξ” is the maximum angle spread at 𝑀𝑆𝑇, determined by the scattering around 𝑀𝑆𝑅. Similarly, 2Ξ”β€² is the maximum angle spread at 𝑀𝑆𝑅, determined by the scattering around 𝑀𝑆𝑇. From Figure 1, it is clear that Ξ”=arcsin(𝑅/𝐷), and Ξ”ξ…ž=arcsin(π‘…ξ…ž/𝐷). Note that geometry of our proposed model includes many existing geometrical models.

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 π‘†π‘˜π‘‡ around the 𝑀𝑆𝑇. For the frequency flat, subchannel between the antenna elements 𝐴𝑝𝑇 and 𝐴𝑙𝑅, β„Žπ‘™π‘(𝑑) 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:β„Žπ‘™π‘ξ‚™(𝑑)=πœ‚π‘‡π‘π‘‡π‘π‘‡ξ“π‘˜=1expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘˜π‘‡+π‘‘π‘†π‘˜π‘‡π΄π‘™π‘…ξ‚+π‘—Ξ¨π‘˜π‘‡+𝑗2πœ‹π‘“π‘˜1𝑑+ξ‚™πœ‚π‘…π‘π‘…π‘π‘…ξ“π‘–=1expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘–π‘…+𝑑𝑆𝑖𝑅𝐴𝑙𝑅+𝑗Ψ𝑖𝑅+𝑗2πœ‹π‘“π‘–2𝑑,(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(𝑑), in which β„Žπ‘™π‘(𝑑) is the element of row 𝑙 and column 𝑝. The π‘‘π‘‹π‘Œ denotes the distance between 𝑋 and π‘Œ, πœ‚π‘‡ and πœ‚π‘… show the respective contributions of scatterers around the 𝑀𝑆𝑇 and 𝑀𝑆𝑅 such that πœ‚π‘‡+πœ‚π‘…=1. 𝑁𝑇 and 𝑁𝑅 are the number of scatterers around the 𝑀𝑆𝑇 and 𝑀𝑆𝑅, respectively. Ξ¨π‘˜π‘‡ and Ψ𝑖𝑅 are the associated phase shifts. Furthermore, as shown in Figure 1, πœ‘π‘˜π‘‡ and πœ‘π‘–π‘… are AoD’s of the waves that impinge on π‘†π‘˜π‘‡ and 𝑆𝑖𝑅. Similarly πœ™π‘˜π‘‡ and πœ™π‘–π‘… are AoA’s of the waves scattered from π‘†π‘˜π‘‡ and 𝑆𝑖𝑅. Note that π‘‘π΄π‘π‘‡π‘†π‘˜π‘‡ and π‘‘π΄π‘žπ‘‡π‘†π‘˜π‘‡ are functions of πœ‘π‘˜π‘‡, whereas π‘‘π‘†π‘˜π‘‡π΄π‘™π‘… and π‘‘π‘†π‘˜π‘‡π΄π‘šπ‘… are functions of πœ™π‘˜π‘‡. Other π‘‘π‘‹π‘Œ can be easily identified from Figure 1. πœ† is the wavelength and frequencies π‘“π‘˜1 and 𝑓𝑖2 are given byπ‘“π‘˜1=𝑓𝑇max𝛼cosπ‘‡βˆ’πœ‘π‘˜π‘‡ξ€Έ+𝑓𝑅max𝛼cosπ‘…βˆ’πœ™π‘˜π‘‡ξ€Έ,𝑓(2)𝑖2=𝑓𝑇max𝛼cosπ‘‡βˆ’πœ‘π‘–π‘…ξ€Έ+𝑓𝑅max𝛼cosπ‘…βˆ’πœ™π‘–π‘…ξ€Έ,(3) where 𝑓𝑇max=𝑣𝑇/πœ† and 𝑓𝑅max=𝑣𝑅/πœ† are the maximum Doppler frequencies caused by the movement of the transmitter and the receiver, respectively. We also assume {Ξ¨π‘˜π‘‡}π‘π‘‡π‘˜=1 and {Ψ𝑖𝑅}𝑁𝑅𝑖=1 are mutually independent and identically distributed (i.i.d) random variables with uniform distributions over [0,2πœ‹). According to Figure 1, while πœ‘π‘–π‘… and πœ™π‘˜π‘‡ are dependent to πœ™π‘–π‘… and πœ‘π‘˜π‘‡, respectively, πœ™π‘–π‘… and πœ‘π‘˜π‘‡ are independent variables. In what follows, we call πœ‘π‘˜π‘‡ the AOD, and πœ™π‘–π‘… 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 β„Žπ‘™π‘(𝑑) and β„Žπ‘šπ‘ž(𝑑) is defined by πœŒπ‘™π‘,π‘šπ‘ž(𝜏)=𝔼[β„Žπ‘™π‘(𝑑)β„Žβˆ—π‘šπ‘ž(𝑑+𝜏)], where 𝔼(β‹…) is the statistical expectation operator and (β‹…)βˆ— denotes complex conjugate operation. Based on independent properties of Ξ¨π‘˜π‘‡ and Ψ𝑖𝑅, it can be asymptotically written byπœŒπ‘™π‘,π‘šπ‘ž(𝜏)=limπ‘π‘‡β†’βˆžπœ‚π‘‡π‘π‘‡π‘π‘‡ξ“π‘˜=1𝔼×expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘˜π‘‡βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘˜π‘‡+π‘‘π‘†π‘˜π‘‡π΄π‘™π‘…βˆ’π‘‘π‘†π‘˜π‘‡π΄π‘šπ‘…ξ‚βˆ’π‘—2πœ‹π‘“π‘˜1πœξ‚‡ξ‚„+limπ‘π‘…β†’βˆžπœ‚π‘…π‘π‘…π‘π‘…ξ“π‘–=1𝔼×expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘–π‘…βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘–π‘…+π‘‘π‘†π‘–π‘…π΄π‘™π‘…βˆ’π‘‘π‘†π‘–π‘…π΄π‘šπ‘…ξ‚βˆ’π‘—2πœ‹π‘“π‘–2𝜏.(4) For large 𝑁𝑇 and 𝑁𝑅, the discrete AoDs, πœ‘π‘˜π‘‡, and the discrete AoAs, πœ™π‘–π‘…, can be replaced with their continuous random variables πœ‘π‘‡ and πœ™π‘… with probability density functions (pdf) 𝑓𝑀𝑆𝑇(πœ‘π‘‡) and 𝑓𝑀𝑆𝑅(πœ™π‘…), respectively. Therefore, (4) can be reduced to the following integral form:πœŒπ‘™π‘,π‘šπ‘ž(𝜏)=πœ‚π‘‡ξ€œπœ‹βˆ’πœ‹ξ‚†expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘‡βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘‡+π‘‘π‘†π‘‡π΄π‘™π‘…βˆ’π‘‘π‘†π‘‡π΄π‘šπ‘…ξ‚βˆ’π‘—2πœ‹π‘“1πœξ‚‡π‘“π‘€π‘†π‘‡ξ€·πœ‘π‘‡ξ€Έπ‘‘πœ‘π‘‡+πœ‚π‘…ξ€œπœ‹βˆ’πœ‹ξ‚†expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘…βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘…+π‘‘π‘†π‘…π΄π‘™π‘…βˆ’π‘‘π‘†π‘…π΄π‘šπ‘…ξ‚βˆ’π‘—2πœ‹π‘“2πœξ‚‡π‘“π‘€π‘†π‘…ξ€·πœ™π‘…ξ€Έπ‘‘πœ™π‘…,(5) where 𝑓1 and 𝑓2 are the continuous form of π‘“π‘˜1 and 𝑓𝑖2 in (2) and (3), respectively. All of the π‘‘π‘‹π‘Œβ€™s in first integral of (5) depend on πœ‘π‘‡ and in the second integral depend on πœ™π‘….

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:π‘‘π΄π‘π‘‡π‘†π‘‡βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘‡β‰ˆβˆ’π›Ώπ‘‡π‘π‘žξ€·π›½cosπ‘‡βˆ’πœ‘π‘‡ξ€Έ,π‘‘π‘†π‘‡π΄π‘™π‘…βˆ’π‘‘π‘†π‘‡π΄π‘šπ‘…β‰ˆβˆ’π›Ώπ‘…π‘™π‘šξ€·π›½cosπ‘…βˆ’πœ™π‘‡ξ€Έ,π‘‘π΄π‘π‘‡π‘†π‘…βˆ’π‘‘π΄π‘žπ‘‡π‘†π‘…β‰ˆβˆ’π›Ώπ‘‡π‘π‘žξ€·π›½cosπ‘‡βˆ’πœ‘π‘…ξ€Έ,π‘‘π‘†π‘…π΄π‘™π‘…βˆ’π‘‘π‘†π‘…π΄π‘šπ‘…β‰ˆβˆ’π›Ώπ‘…π‘™π‘šξ€·π›½cosπ‘…βˆ’πœ™π‘…ξ€Έ.(6) Now we apply the law of sines and obtain the following identities:π·ξ€·πœ™sinπ‘‡βˆ’πœ‘π‘‡ξ€Έ=𝑅′sinπœ‹βˆ’πœ™π‘‡ξ€Έ,π·ξ€·πœ™sinπ‘…βˆ’πœ‘π‘…ξ€Έ=π‘…ξ€·πœ‘sin𝑅.(7) Based on the assumption max{𝑅,𝑅′}β‰ͺ𝐷, we conclude that Ξ”β‰ˆπ‘…/𝐷, and Ξ”β€²β‰ˆπ‘…β€²/𝐷. This observation, together with sinπœ–β‰ˆπœ– when πœ– is small, considering πœ™π‘‡ is almost πœ‹ and πœ‘π‘… is almost 0, allows us to derive the following approximations from (7):πœ™π‘‡β‰ˆπœ‹βˆ’Ξ”ξ…žsinπœ‘π‘‡,πœ‘π‘…β‰ˆΞ”sinπœ™π‘….(8) Furthermore, using sinπœ–β‰ˆπœ– and cosπœ–β‰ˆ1 when πœ– is small, together with (8), the following approximations are derived:𝛽cosπ‘…βˆ’πœ™π‘‡ξ€Έβ‰ˆβˆ’cos𝛽𝑅+Ξ”ξ…žsin𝛽𝑅sinπœ‘π‘‡ξ€·π›Ό,(9)cosπ‘…βˆ’πœ™π‘‡ξ€Έβ‰ˆβˆ’cos𝛼𝑅+Ξ”ξ…žsin𝛼𝑅sinπœ‘π‘‡ξ€·π›½,(10)cosπ‘‡βˆ’πœ‘π‘…ξ€Έβ‰ˆcos𝛽𝑇+Ξ”sin𝛽𝑇sinπœ™π‘…ξ€·π›Ό,(11)cosπ‘‡βˆ’πœ‘π‘…ξ€Έβ‰ˆcos𝛼𝑇+Ξ”sin𝛼𝑇sinπœ™π‘….(12)

Now, by substituting (10) and (12) to continuous form of (2) and (3), respectively, the following approximations are derived:𝑓1β‰ˆπ‘“π‘‡max𝛼cosπ‘‡βˆ’πœ‘π‘‡ξ€Έβˆ’π‘“π‘…maxcos𝛼𝑅+𝑓𝑅maxΞ”ξ…žsinπœ‘π‘‡sin𝛼𝑅,𝑓2β‰ˆπ‘“π‘‡maxcos𝛼𝑇+𝑓𝑇maxΞ”sinπœ™π‘…sin𝛼𝑇+𝑓𝑅max𝛼cosπ‘…βˆ’πœ™π‘…ξ€Έ.(13)

For any given 𝑓𝑀𝑆𝑇(β‹…) and 𝑓𝑀𝑆𝑅(β‹…), 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′}β‰ͺ𝐷 and min{𝑅,𝑅′}≫max{𝛿𝑅,𝛿𝑇}, by plugging (6), (9) and (11) into (5), equation (5) is approximated byπœŒπ‘™π‘,π‘šπ‘ž(𝜏)β‰ˆπœ‚π‘‡ξ€œπœ‹βˆ’πœ‹ξ‚†π‘—exp2πœ‹πœ†ξ€·π›Ώπ‘‡π‘π‘žξ€·π›½cosπ‘‡βˆ’πœ‘π‘‡ξ€Έ+π›Ώπ‘…π‘™π‘šΓ—ξ€·βˆ’cos𝛽𝑅+Ξ”ξ…žsin𝛽𝑅sinπœ‘π‘‡ξ€Έξ€Έβˆ’π‘—2πœ‹π‘“1πœξ‚‡π‘“π‘€π‘†π‘‡ξ€·πœ‘π‘‡ξ€Έπ‘‘πœ‘π‘‡+πœ‚π‘…ξ€œπœ‹βˆ’πœ‹ξ‚†π‘—exp2πœ‹πœ†ξ€·π›Ώπ‘‡π‘π‘žξ€·cos𝛽𝑇+Ξ”sin𝛽𝑇sinπœ™π‘…ξ€Έ+π›Ώπ‘…π‘™π‘šξ€·π›½cosπ‘…βˆ’πœ™π‘…ξ€Έξ€Έβˆ’π‘—2πœ‹π‘“2πœξ‚‡π‘“π‘€π‘†π‘…ξ€·πœ™π‘…ξ€Έπ‘‘πœ™π‘….(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(πœƒ)=2πœ‹πΌ0[](π‘˜)expπ‘˜cos(πœƒβˆ’πœ‡),(15) where 𝐼0(β‹…) is the zeroth-order modified Bessel function of the first kind, πœ‡βˆˆ[βˆ’πœ‹,πœ‹) is the mean angle of scatterers’ distribution on the ring, and π‘˜ controls the spread of scatterers around the mean. When π‘˜=0, 𝑝(πœƒ)=1/(2πœ‹) is a uniform distribution yielding 2D isotropic scattering. As π‘˜ increases, the scatterers become more clustered around angle πœ‡ and the scattering becomes increasingly nonisotropic. Therefor, the von Mises pdf of AOD and AOA is given by 𝑓𝑀𝑆𝑇(πœ‘π‘‡)=exp[π‘˜π‘‡cos(πœ‘π‘‡βˆ’πœ‡π‘‡)]/(2πœ‹πΌ0(π‘˜π‘‡)) and 𝑓𝑀𝑆𝑅(πœ™π‘…)=exp[π‘˜π‘…cos(πœ™π‘…βˆ’πœ‡π‘…)]/(2πœ‹πΌ0(π‘˜π‘…)), respectively.

From [[19], eq. 3.338], we haveξ€œπœ‹βˆ’πœ‹exp(π‘₯sinπœƒ+𝑦cosπœƒ)π‘‘πœƒ=2πœ‹πΌ0ξ‚€βˆšπ‘₯2+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)). πœŒπ‘™π‘,π‘šπ‘žπœ‚(𝜏)β‰ˆπ‘‡πΌ0ξ€·π‘˜π‘‡ξ€Έξ‚ƒexpβˆ’π‘—2πœ‹πœ†π›Ώπ‘…π‘™π‘šcos𝛽𝑅+𝑗2πœ‹π‘“π‘…max𝜏cos𝛼𝑅×𝐼0π‘˜ξƒ©ξƒ―2π‘‡βˆ’ξ‚€2πœ‹πœ†ξ‚2π›Ώπ‘…π‘™π‘šΞ”ξ…žsinπ›½π‘…ξ€·π›Ώπ‘…π‘™π‘šΞ”ξ…žsin𝛽𝑅+2π›Ώπ‘‡π‘π‘žsinπ›½π‘‡ξ€Έβˆ’(2πœ‹)2πœ†π›Ώπ‘‡π‘π‘žξƒ©π›Ώπ‘‡π‘π‘žπœ†βˆ’2𝑓𝑇maxξ€·π›Όπœcosπ‘‡βˆ’π›½π‘‡ξ€Έξƒͺβˆ’ξ‚€2πœ‹π‘“π‘‡maxπœξ‚2+2(2πœ‹)2πœ†π›Ώπ‘…π‘™π‘šΞ”ξ…žπœsin𝛽𝑅𝑓𝑇maxsin𝛼𝑇+Ξ”ξ…žπ‘“π‘…maxsinπ›Όπ‘…ξ‚βˆ’(2πœ‹)2×𝑓𝑅maxπœΞ”ξ…žsin𝛼𝑅𝑓𝑅maxπœΞ”ξ…žsinπ›Όπ‘…βˆ’2πœ†π›Ώπ‘‡π‘π‘žsin𝛽𝑇+2𝑓𝑇max𝜏sin𝛼𝑇+𝑗2π‘˜π‘‡ξ‚ƒ2πœ‹πœ†π›Ώπ‘‡π‘π‘žξ€·π›½cosπ‘‡βˆ’πœ‡π‘‡ξ€Έβˆ’2πœ‹π‘“π‘‡maxξ€·π›Όπœcosπ‘‡βˆ’πœ‡π‘‡ξ€Έ+2πœ‹πœ†π›Ώπ‘…π‘™π‘šΞ”ξ…žsin𝛽𝑅sinπœ‡π‘‡βˆ’2πœ‹π‘“π‘…maxπœΞ”ξ…žsin𝛼𝑅sinπœ‡π‘‡ξ‚„ξ‚‡1/2ξ‚Ά+πœ‚π‘…πΌ0ξ€·π‘˜π‘…ξ€Έξ‚ƒπ‘—exp2πœ‹πœ†π›Ώπ‘‡π‘π‘žcosπ›½π‘‡βˆ’π‘—2πœ‹π‘“π‘‡max𝜏cos𝛼𝑇×𝐼0π‘˜ξ‚΅ξ‚»2π‘…βˆ’ξ‚€2πœ‹πœ†ξ‚2π›Ώπ‘‡π‘π‘žΞ”sinπ›½π‘‡ξ€·π›Ώπ‘‡π‘π‘žΞ”sin𝛽𝑇+2π›Ώπ‘…π‘™π‘šsinπ›½π‘…ξ€Έβˆ’(2πœ‹)2πœ†π›Ώπ‘…π‘™π‘šξƒ©π›Ώπ‘…π‘™π‘šπœ†βˆ’2𝑓𝑅maxξ€·π›Όπœcosπ‘…βˆ’π›½π‘…ξ€Έξƒͺβˆ’ξ‚€2πœ‹π‘“π‘…maxπœξ‚2+2(2πœ‹)2πœ†π›Ώπ‘‡π‘π‘žΞ”πœΓ—sin𝛽𝑇𝑓𝑅maxsin𝛼𝑅+Δ𝑓𝑇maxsinπ›Όπ‘‡ξ‚βˆ’(2πœ‹)2𝑓𝑇maxπœΞ”sin𝛼𝑇𝑓𝑇maxπœΞ”sinπ›Όπ‘‡βˆ’2πœ†π›Ώπ‘…π‘™π‘šsin𝛽𝑅+2𝑓𝑅max𝜏sin𝛼𝑅+𝑗2π‘˜π‘…ξ‚ƒ2πœ‹πœ†π›Ώπ‘…π‘™π‘šξ€·π›½cosπ‘…βˆ’πœ‡π‘…ξ€Έβˆ’2πœ‹π‘“π‘…maxξ€·π›Όπœcosπ‘…βˆ’πœ‡π‘…ξ€Έ+2πœ‹πœ†π›Ώπ‘‡π‘π‘žΞ”sin𝛽𝑇sinπœ‡π‘…βˆ’2πœ‹π‘“π‘‡maxπœΞ”sin𝛼𝑇sinπœ‡π‘…ξ‚„ξ‚Ό1/2ξƒͺ.(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 (π‘˜π‘‡=π‘˜π‘…=0), the STCCF of our reference model reduces to STCCF of MGTR model in isotropic environment [20] asπœŒπ‘™π‘,π‘šπ‘ž(𝜏)β‰ˆπœ‚π‘‡ξ‚ƒexpβˆ’π‘—2πœ‹πœ†π›Ώπ‘…π‘™π‘šcos𝛽𝑅+𝑗2πœ‹π‘“π‘…max𝜏cos𝛼𝑅×𝐼0βˆ’ξ‚€ξ‚΅ξ‚»2πœ‹πœ†ξ‚2π›Ώπ‘…π‘™π‘šΞ”ξ…žsinπ›½π‘…ξ€·π›Ώπ‘…π‘™π‘šΞ”ξ…žsin𝛽𝑅+2π›Ώπ‘‡π‘π‘žsinπ›½π‘‡ξ€Έβˆ’(2πœ‹)2πœ†π›Ώπ‘‡π‘π‘žξƒ©π›Ώπ‘‡π‘π‘žπœ†βˆ’2𝑓𝑇maxξ€·π›Όπœcosπ‘‡βˆ’π›½π‘‡ξ€Έξƒͺβˆ’ξ‚€2πœ‹π‘“π‘‡maxπœξ‚2+2(2πœ‹)2πœ†π›Ώπ‘…π‘™π‘šΞ”ξ…žπœsin𝛽𝑅𝑓𝑇maxsin𝛼𝑇+Ξ”ξ…žπ‘“π‘…maxsinπ›Όπ‘…ξ‚βˆ’(2πœ‹)2𝑓𝑅maxπœΞ”ξ…žsin𝛼𝑅𝑓𝑅maxπœΞ”ξ…žsinπ›Όπ‘…βˆ’2πœ†π›Ώπ‘‡π‘π‘žsin𝛽𝑇+2𝑓𝑇max𝜏sin𝛼𝑇1/2ξƒͺ+πœ‚π‘…ξ‚ƒπ‘—exp2πœ‹πœ†π›Ώπ‘‡π‘π‘žcosπ›½π‘‡βˆ’π‘—2πœ‹π‘“π‘‡max𝜏cos𝛼𝑇×𝐼0βˆ’ξ‚€ξ‚΅ξ‚»2πœ‹πœ†ξ‚2π›Ώπ‘‡π‘π‘žΞ”sinπ›½π‘‡ξ€·π›Ώπ‘‡π‘π‘žΞ”sin𝛽𝑇+2π›Ώπ‘…π‘™π‘šsinπ›½π‘…ξ€Έβˆ’(2πœ‹)2πœ†π›Ώπ‘…π‘™π‘šξƒ©π›Ώπ‘…π‘™π‘šπœ†βˆ’2𝑓𝑅maxξ€·π›Όπœcosπ‘…βˆ’π›½π‘…ξ€Έξƒͺβˆ’ξ‚€2πœ‹π‘“π‘…maxπœξ‚2+2(2πœ‹)2πœ†π›Ώπ‘‡π‘π‘žΞ”πœsin𝛽𝑇𝑓𝑅maxsin𝛼𝑅+Δ𝑓𝑇maxsinπ›Όπ‘‡ξ‚βˆ’(2πœ‹)2𝑓𝑇maxπœΞ”sin𝛼𝑇𝑓𝑇maxπœΞ”sinπ›Όπ‘‡βˆ’2πœ†π›Ώπ‘…π‘™π‘šsin𝛽𝑅+2𝑓𝑅max𝜏sin𝛼𝑅1/2ξƒͺ.(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 (πœ‚π‘‡=0) and stationary 𝑀𝑆𝑇 (𝑓𝑇max=0), (17) is simplified to STCCF of the conventional β€œone-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 (πœ‚π‘‡=0), stationary 𝑀𝑆𝑇 (𝑓𝑇max=0) and, with 𝑙=π‘š and 𝑝=π‘ž, our reference model’s STCCF is simplified to conventional β€œone-ring” model for SISO B-to-M communication channel. This reduces (17) to the well-known Clarke’s temporal correlation function, that is, 𝐽0(2πœ‹π‘“π‘…max𝜏) [21], where 𝐽0(β‹…) 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):ξβ„Žπ‘™π‘ξ‚™(𝑑)=πœ‚π‘‡π‘π‘‡π‘π‘‡ξ“π‘˜=1expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘˜π‘‡+π‘‘π‘†π‘˜π‘‡π΄π‘™π‘…ξ‚+π‘—Ξ¨π‘˜π‘‡+𝑗2πœ‹π‘“π‘˜1𝑑+ξ‚™πœ‚π‘…π‘π‘…π‘π‘…ξ“π‘–=1expβˆ’π‘—2πœ‹πœ†ξ‚€π‘‘π΄π‘π‘‡π‘†π‘–π‘…+𝑑𝑆𝑖𝑅𝐴𝑙𝑅+𝑗Ψ𝑖𝑅+𝑗2πœ‹π‘“π‘–2𝑑,(19) where the parameters of above equation are defined in Section 2.2. In contrast to the reference model, the discrete AODs πœ‘π‘˜π‘‡ and AOAs πœ™π‘–π‘… are now constant, which will be determined in Section 5. The phases Ξ¨π‘˜π‘‡ and Ψ𝑖𝑅 are still i.i.d. random variables, each with uniform distribution on the interval [0,2πœ‹). Hence, ξβ„Žπ‘™π‘(𝑑) represents a stochastic process. The STCCF between ξβ„Žπ‘™π‘(𝑑) and ξβ„Žπ‘šπ‘ž(𝑑) is defined as Μ‚πœŒπ‘™π‘,π‘šπ‘žξβ„Ž(𝜏)=𝔼[π‘™π‘ξβ„Ž(𝑑)βˆ—π‘šπ‘ž(𝑑+𝜏)], where (β‹…)βˆ— denotes the complex conjugate operation, and 𝔼(β‹…) is the statistical expectation operator, which applies to the random phases Ξ¨π‘˜π‘‡ 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), asΜ‚πœŒπ‘™π‘,π‘šπ‘žπœ‚(𝜏)β‰ˆπ‘‡π‘π‘‡π‘π‘‡ξ“π‘˜=1×𝑗exp2πœ‹πœ†ξ€·π›Ώπ‘‡π‘π‘žξ€·π›½cosπ‘‡βˆ’πœ‘π‘˜π‘‡ξ€Έ+π›Ώπ‘…π‘™π‘šΓ—ξ€·βˆ’cos𝛽𝑅+Ξ”ξ…žsin𝛽𝑅sinπœ‘π‘˜π‘‡ξ€Έξ€Έβˆ’π‘—2πœ‹π‘“π‘˜1πœξ‚‡+πœ‚π‘…π‘π‘…π‘π‘…ξ“π‘–=1𝑗exp2πœ‹πœ†ξ€·π›Ώπ‘‡π‘π‘žξ€·cos𝛽𝑇+Ξ”sin𝛽𝑇sinπœ™π‘–π‘…ξ€Έ+π›Ώπ‘…π‘™π‘šξ€·π›½cosπ‘…βˆ’πœ™π‘–π‘…ξ€Έξ€Έβˆ’π‘—2πœ‹π‘“π‘–2πœξ‚‡,(20) whereπ‘“π‘˜1β‰ˆπ‘“π‘‡max𝛼cosπ‘‡βˆ’πœ‘π‘˜π‘‡ξ€Έβˆ’π‘“π‘…maxcos𝛼𝑅+𝑓𝑅maxΞ”ξ…žsinπœ‘π‘˜π‘‡sin𝛼𝑅𝑓𝑖2β‰ˆπ‘“π‘‡maxcos𝛼𝑇+𝑓𝑇maxΞ”sinπœ™π‘–π‘…sin𝛼𝑇+𝑓𝑅max𝛼cosπ‘…βˆ’πœ™π‘–π‘…ξ€Έ.(21)

In the following section we introduce two methods for determining the constant discrete AODs πœ‘π‘˜π‘‡ and AOAs πœ™π‘–π‘….

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 πœ‘π‘˜π‘‡ and AOAs πœ™π‘–π‘… are determined by [22]:πœ‘π‘˜π‘‡=2πœ‹π‘π‘‡ξ‚€1π‘˜βˆ’2,π‘˜=1,2,…,𝑁𝑇,πœ™π‘–π‘…=2πœ‹π‘π‘…ξ‚€1π‘–βˆ’2,𝑖=1,2,…,𝑁𝑅.(22) Therefore, in the statistical simulation model only phases Ξ¨π‘˜π‘‡ and Ψ𝑖𝑅 are random parameters. They are i.i.d. random variables uniformly distributed over [0,2πœ‹).

5.2. 𝐿𝑝-Norm Method

When the AODs πœ‘π‘˜π‘‡ and AOAs πœ™π‘–π‘… 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 πœ‘π‘˜π‘‡ and AOAs πœ™π‘–π‘… are determined by minimizing the following error norm [22]:𝐸𝜌(𝑝)𝑙𝑝,π‘šπ‘ž=ξ‚»1𝜏maxξ€œπœmax0||πœŒπ‘™π‘,π‘šπ‘ž(𝜏)βˆ’Μ‚πœŒπ‘™π‘,π‘šπ‘ž||(𝜏)π‘ξ‚Όπ‘‘πœ1/𝑝,(23) where 𝑝=1,2,…, πœŒπ‘™π‘,π‘šπ‘ž(𝜏) and Μ‚πœŒπ‘™π‘,π‘šπ‘ž(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 Ξ¨π‘˜π‘‡ and Ψ𝑖𝑅 are random parameters. They are i.i.d random variables uniformly distributed over [0,2πœ‹).

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œ‹/2. At the transmitter side, the angle of motion 𝛼𝑇 was set to πœ‹/4, while the receiver was moving at an angle of 𝛼𝑅=0. Identical maximum Doppler frequencies 𝑓𝑇max=𝑓𝑅max=91Hz were assumed, and the wavelength πœ† was set to πœ†=0.15m (according to [10, 11]). Furthermore, the other parameters have their quantity as Ξ”=πœ‹/3, Ξ”β€²=πœ‹/6, πœ‡π‘‡=5πœ‹/8, πœ‡π‘…=0, and πœ‚π‘…=0.2, according to Table I of [17].

6.1. The MEDS Simulation Model

Such as mentioned before, the model parameters πœ‘π‘˜π‘‡ and πœ™π‘–π‘… have been determined by the MEDS method, since we assume isotropic scattering around both the transmitter and the receiver (π‘˜π‘‡=π‘˜π‘…=0). Note that the mismatch criteria in the following simulation is Relative Error and is set to 10βˆ’3. The relative error is defined byπœ–πœŒπ‘™π‘,π‘šπ‘ž=1/𝜏maxξ€Έβˆ«πœmax0||πœŒπ‘™π‘,π‘šπ‘ž(𝜏)βˆ’Μ‚πœŒπ‘™π‘,π‘šπ‘ž||(𝜏)2ξ‚‡π‘‘πœ1/21/𝜏maxξ€Έβˆ«πœmax0||πœŒπ‘™π‘,π‘šπ‘ž||(𝜏)2ξ‚‡π‘‘πœ1/2,(24) where πœŒπ‘™π‘,π‘šπ‘ž(𝜏) and Μ‚πœŒπ‘™π‘,π‘šπ‘ž(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 (Μ‚πœŒπ‘™π‘,π‘šπ‘ž(𝜏)) with the STCCF of the theoretical isotropic reference model (πœŒπ‘™π‘,π‘šπ‘ž(𝜏)) for π›Ώπ‘‡π‘π‘ž=π›Ώπ‘…π‘™π‘š=1πœ†. 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.

6.2. The 𝐿𝑝-Norm Simulation Model

For performance evaluation of 𝐿𝑝-Norm simulation model, first we determine the optimum parameters πœ‘π‘˜π‘‡ and πœ™π‘–π‘… by minimizing the error norm defined in (23). Note that in the following simulation scenarios, we assume π‘˜π‘‡=0.5 and π‘˜π‘…=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 π›Ώπ‘‡π‘π‘ž=π›Ώπ‘…π‘™π‘š=1πœ† 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.

Note that the number displayed on the Figures 4–7 is the minimized relative error of 𝐿𝑝-Norm method and defined as𝐸relativeerror=𝜌(2)𝑙𝑝,π‘šπ‘žξ‚†ξ€·1/𝜏maxξ€Έβˆ«πœmax0||πœŒπ‘™π‘,π‘šπ‘ž||(𝜏)2ξ‚‡π‘‘πœ1/2.(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.