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Hindawi Publishing Corporation EURASIP Journal on Advances in Signal Processing Volume 2011, Article ID 892871, 13 pages doi:10.1155/2011/892871 Research Article A Geometrical Three-Ring-Based Model for MIMO Mobile-to-Mobile Fading Channels in Cooperative Networks Batool Talha and Matthias P¨ tzold a Faculty of Engineering and Science, University of Agder, Servicebox 509, 4898 Grimstad, Norway Correspondence should be addressed to Batool Talha, batool.talha@uia.no Received 2 June 2010; Revised 4 October 2010; Accepted 2 January 2011 Academic Editor: Francesco Verde Copyright © 2011 B. Talha and M. P¨ tzold. This is an open access article distributed under the Creative Commons Attribution a License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. This paper deals with the modeling and analysis of narrowband multiple-input multiple-output (MIMO) mobile-to-mobile (M2M) fading channels in relay-based cooperative networks. In the transmission links from the source mobile station to the destination mobile station via the mobile relay, non-line-of-sight (NLOS) propagation conditions are taken into account. A stochastic narrowband MIMO M2M reference channel model is derived from the geometrical three-ring scattering model, where it is assumed that an inﬁnite number of local scatterers surround the source mobile station, the mobile relay, and the destination mobile station. The complex channel gains associated with the new reference channel model are derived, and their temporal as well as spatial correlation properties are explored. General analytical solutions are obtained for the four-dimensional (4D) space-time cross-correlation function (CCF), the three-dimensional (3D) spatial CCF, the two-dimensional (2D) source (relay, destination) correlation function (CF), and the temporal autocorrelation function (ACF). Exact closed-form expressions for diﬀerent CFs under isotropic as well as nonisotropic scattering conditions are provided in this article. A stochastic simulation model is then drawn from the reference model. It is shown that the CCFs of the simulation model closely approximate the corresponding CCFs of the reference model. The developed channel simulator is not only important for the development of future MIMO M2M cooperative communication systems, but also for analyzing the dynamic behavior of the MIMO M2M channel capacity. yet eﬃcient M2M fading channel models, providing us with 1. Introduction a detailed knowledge about the statistical characterization of Recent attempts to combat multipath fading eﬀects along M2M channels. with providing increased mobility support have resulted in The idea of introducing M2M communication in nonco- the emergence of M2M communication systems in cooper- operative networks can be traced back to the work of Akki ative networks. The use of cooperative diversity protocols and Haber [5, 6], which deals with the study of the statistical [1–4] improves the transmission link quality and the end- properties of narrowband single-input single-output (SISO) to-end system throughput, whereas M2M communication, M2M fading channels under non-line-of-sight (NLOS) on the other hand, expands the network range (coverage propagation conditions. Several papers dealing with M2M area). The fundamental idea of cooperative networks is communication in cooperative networks can be found in the to allow mobile stations in the network to relay signals recent literature [7–9]. In various studies regarding M2M to the ﬁnal destination or to other mobile stations acting fading channels in relay-based cooperative networks under as relays. The development and performance investigation NLOS propagation conditions, it has been shown that a of such seemingly straightforward cooperative networks double Rayleigh process is an unsophisticated but still a require a thorough understanding of the M2M fading chan- well-suited statistical channel model for such channels [10, nel characteristics. For this reason, there is a need for simple 11]. Besides, the credit of reporting the temporal ACF of
2 EURASIP Journal on Advances in Signal Processing fading channels in amplify-and-forward relay systems goes to and simulation approaches for such channels in amplify- Patel et al. [11]. The analysis of experimental measurement and-forward relay-type cooperative networks. Additionally, data for outdoor-to-indoor M2M fading channels included there was not a single M2M channel model for cooperative in [12] veriﬁes the existence of double Rayleigh processes networks available in the literature, which assumes multiple in real-world environments. Talha and P¨tzold [7] have a antennas on the source mobile station, the destination extended the double Rayleigh channel model to the double mobile station, or the mobile relays. This gap in the Rice channel model for line-of-sight (LOS) propagation research propelled us to introduce a geometry-based model environments. Furthermore, a variety of other realistic M2M for MIMO M2M channels in relay-based systems. The fading channel models based on the multiple scattering scattering environment around the source mobile station, concept [13] are available in the literature for both NLOS the mobile relay, and the destination mobile station are and LOS propagation environments [9, 14]. The M2M modeled by a geometrical three-ring scattering model. fading channel models for relay-based cooperative networks The advanced geometrical three-ring scattering model is proposed to date are for narrowband SISO fading channels. an extension of the geometrical two-ring scattering model Meaning thereby, the source mobile station, the mobile relay, presented in [30], where the source mobile station and and the destination mobile station are equipped with only the destination mobile station are surrounded by rings one antenna. However, it is a well-established fact that the of scatterers. However, in the suggested extension of the gains in terms of channel capacity are larger for MIMO two-ring model to the three-ring model, we have a sep- channels as compared to SISO channels [15, 16]. This thus arate ring of scatterers around the mobile relay in addi- calls for an extension of SISO M2M channel models to tion to a ring around each source mobile station and MIMO M2M channel models, since such models facilitate destination mobile station. For simplicity, the reference investigations pertaining to the channel capacity and the model introduced in this article caters for MIMO M2M system performance of cooperative networks with multiple fading channels under NLOS propagation conditions only. antenna models. Moreover, it is assumed that the direct transmission link Another area that requires further attention is the between the source mobile station and the destination development of simulation models for MIMO M2M fading mobile station is blocked by obstacles. Since geometry- channels in cooperative networks. Some techniques for based MIMO channel models are usually characterized by simulating narrowband SISO M2M fading channels in their temporal as well as spatial correlation properties, we noncooperative networks can be found in [17]. Various explore the correlation properties of our devised three- studies have revealed that geometrical channel models ring-based model here. It is noteworthy that while deriving are a good starting point for deriving simulation models the temporal ACF of the relay links, the authors of [11] for MIMO channels. Quite a lot of narrowband MIMO took into consideration such propagation scenarios where channel models based on geometrical scattering models for a stationary base station (BS) acts either as a source (i.e., isotropic environments have been developed so far [18–25]. transmitter) or a relay. It is usually supposed that the BS The design of geometry-based MIMO channel models for is elevated and unobstructed. It is further believed that the nonisotropic scattering conditions is addressed in [26, 27], BS is not surrounded by local scatterers. This assumption of the elevated BS makes [11] diﬀerent from our work. whereas wideband MIMO channel models are discussed in [28, 29]. The common feature in the works [18–29] Additionally, we derive a stochastic simulation model from is that they model MIMO ﬁxed-to-mobile (F2M) and/or the developed reference model. Finally, we discuss the ﬁxed-to-ﬁxed (F2F) channels. The geometrical two-ring- nonisotropic scattering scenario along with the isotropic one based model for MIMO F2M channels originally proposed as a special case and present closed-form expressions for the in [21] was extended to a narrowband MIMO M2M channel correlation functions of the reference as well as simulation model by P¨tzold et al. [30]. Zaji´ and St¨ ber [31] have a c u models. reported on MIMO M2M reference and simulation models This article has the following structure. Section 2 intro- under LOS propagation conditions. The geometrical street duces brieﬂy the geometrical three-ring scattering model model [32] and the geometrical T-junction model [33] describing the transmission link from the source mobile for MIMO M2M fading channels are worth mentioning. station to the destination mobile station via the mobile All the geometry-based channel models mentioned to this relay. Based on the geometrical three-ring scattering model, point in the current section are 2D channel models. There we develop the reference model for MIMO M2M fading are also Zaji´ and St¨ ber [34] who have successfully c u channels and study its correlation properties in Section 3. In expanded 2D MIMO M2M channel models to 3D models Section 6, we derive the stochastic simulation model from based on geometrical cylinders. Nonetheless, to the best the developed reference model. Section 4 deals with the of the authors’ knowledge, geometrical channel models derivation of closed-form expressions for the correlation for MIMO M2M communication systems in cooperative functions describing the reference model under nonisotropic networks are an unexplored area. This in turn results in scattering conditions. Section 5 shows the accuracy of the a lack of proper reference and simulation models derived stochastic simulation model by comparing its statistical from such geometrical models for MIMO M2M fading properties with those of the reference model. In this section, channels. we also conﬁrm the validity of the closed-form expressions Motivated by the need for proper MIMO M2M fading obtained in Section 4. Finally, concluding remarks are given channel models, we are addressing in this article modeling in Section 7.
EURASIP Journal on Advances in Signal Processing 3 2. The Geometrical Three-Ring Model 3. The Reference Model 3.1. Derivation of the Reference Model. In this section, we In this section, we extend the geometrical two-ring scattering develop a reference model for MIMO M2M fading chan- model proposed in [30] to a geometrical three-ring scatter- nels in cooperative networks using the geometrical three- ing model for narrowband MIMO M2M fading channels ring scattering model shown in Figure 1. Ignoring for the in amplify-and-forward relay-type cooperative networks. moment the geometrical details, Figure 1 can be simpliﬁed to For ease of analysis, we have considered an elementary Figure 2, in order to understand the overall MIMO channel 2 × 2 × 2 antenna conﬁguration, meaning thereby, the from the source mobile station to the destination mobile source mobile station, the mobile relay, and the destination station via the mobile relay. Figure 2 shows that the complete mobile station are equipped with two antennas each. For system can be separated into two 2 × 2 MIMO subsystems. simplicity, NLOS propagation conditions have been taken One of the MIMO subsystems (comprising the source mobile into account in all the transmission links. It is further station and mobile relay) is denoted by the S-R MIMO assumed that there is no direct transmission link from the subsystem. While the other MIMO subsystem (consisting source mobile station to the destination mobile station. of the mobile relay and the destination mobile station) is termed as the R-D MIMO subsystem. The input-output Due to high path loss, the contribution of signal power relationship of the S-R MIMO subsystem can be expressed from remote scatterers to the total received power is usually as negligible. In this context, the recommended three-ring scattering model only accommodates local scattering. A X(t ) = HS-R (t )S(t ) + NR (t ), (1) total number of M local scatterers, that is, S(m) (m = S 1, 2, . . . , M ) are positioned on a ring of radius RS around the T where X(t ) = [X (1) (t ) X (2) (t )] is a 2 × 1 received signal source mobile station, whereas N local scatterers S(n) (n = T vector at the mobile relay, S(t ) = [S(1) (t ) S(2) (t )] is a 2 × 1 D 1, 2, . . . , N ) lie around the destination mobile station on signal vector transmitted by the source mobile station, and a separate ring of radius RD . Besides, the local scatterers T (1) (2) NR (t ) = [NR (t ) NR (t )] is a 2 × 1 additive white Gaussian S(k) (k = 1, 2, . . . , K ) and S(l) (l = 1, 2, . . . , L) are located on a R R noise (AWGN) vector. In (1), HS-R (t ) is a 2 × 2 channel third ring of radius RR around the mobile relay. The number matrix, which models the M2M fading channel between the of local scatterers around the mobile relay is K = L. It should source mobile station and the mobile relay. The channel be pointed out here that S(k) = S(l) for k = l. Throughout R R matrix HS-R (t ) can be expressed as this paper, the subscripts S, R, and D represent the source ⎛ ⎞ mobile station, the mobile relay, and the destination mobile h(11) (t ) h(12) (t ) S-R S-R HS-R (t ) = ⎝ ⎠. station, respectively. As can be seen from Figure 1, the symbol (2) (21) (22) φSm) denotes the angle of departure (AOD) of the mth ( hS-R (t ) hS-R (t ) transmitted wave seen at the source mobile station, whereas (iq) φD ) represents the angle of arrival (AOA) of the nth received (n Here, each element hS-R (t ) (i, q = 1, 2) of the channel matrix represents the diﬀuse component of the channel describing wave at the destination mobile station. Furthermore, the (k ) (l) the transmission link from the source mobile station antenna symbols φS-R and φR-D correspond to the AOA of the kth (q ) element AS to the mobile relay antenna element A(i) . Con- received wave and the AOD of the lth transmitted wave at R sidering the geometrical three-ring scattering model shown the mobile relay, respectively. The mobile relay is positioned in Figure 1, it can be observed that the mth homogeneous at a distance DS-R and an angle γS with respect to the (q ) source mobile station. While the location of the mobile relay plane wave emitted from AS , ﬁrst encounters the local seen from the destination mobile station can be speciﬁed ( m) scatterers SS around the source mobile station. Moreover, by the distance DR-D and the angle γD . In addition, the ( i) before impinging on AR , the plane wave is captured by source mobile station and the destination mobile station the local scatterers S(k) around the mobile relay. It is worth are a distance DS-D apart from each other. It is expected R mentioning here that the reference model is based on the that the inequalities max{RS , RR } DS-R , max{RR , RD } assumption that the number of local scatterers, M and K , DR-D , and max{RS , RD } DS-D hold. The interelement around the source mobile station and the mobile relay is spacings at the source mobile station, the mobile relay, and (11) inﬁnite. Following [30], the diﬀuse component hS-R (t ) of the the destination mobile station antenna arrays are labeled (1) (1) as δS , δR , and δD , respectively, where it is understood that transmission link from AS to AR can be approximated as these quantities are smaller than the radii RS , RR , and RD , M K that is, max{δS , δR , δD } min{RS , RR , RD }. With respect 1 (m) (k ) (mk ) (11) (mk) hS-R (t ) = lim √ gS-R e j [2π ( fS + fS-R )t+(θS-R +θS-R )] to the x-axis, the symbols βS , βR , and βD describe the tilt M → ∞ MK m=1 k=1 K →∞ angle of the antenna arrays at the source mobile station, the (3) mobile relay, and the destination mobile station, respectively. √ Additionally, it is supposed that the source mobile station (mk) with joint gains 1/ MK and joint phases θS-R caused by (mobile relay, destination mobile station) moves with speed the interaction of the local scatterers S(m) and S(k) . The joint vS (vR , vD ) in the direction determined by the angle of S R (mk) motion αS (αR , αD ). phases θS-R are considered to be independent and identically
4 EURASIP Journal on Advances in Signal Processing Source mobile station Destination mobile station − → − → v v S D δS (1) (1) AS d1m AD (m) δD αS SS (n) φD βD (m) βS φS DS-D αD γD γS dn1 (2) (2) AS AD dmk (n) SD RD RS DS-R DR-D − → (k ) v dln SR R dk1 y (1) AR d1l (l) αR SR βR x (l) φR-D (k ) φS-R Mobile relay δR (2) AR RR Figure 1: The geometrical three-ring scattering model for a 2 × 2 × 2 MIMO M2M channel with local scatterers on rings around the source mobile station, the mobile relay, and the destination mobile station. S(1) (t ) X (1) (t ) R(1) (t ) (11) (11) hS-R (t ) hR-D (t ) (1) (1) AS AD ) ) 2) (t 2) (t (1) AR (1 R (1 -D h S- h R h (2 h (2 R- 1) S- 1) X (2) (t ) S(2) (t ) R(2) (t ) D (t) R( t) (2) (2) AS AD (2) (22) (22) AR hS-R (t ) hR-D (t ) HS-R (t ) HR-D (t ) Destination mobile station Source mobile station Mobile relay Figure 2: A simpliﬁed diagram describing the overall MIMO M2M channel from the source mobile station to the destination mobile station via the mobile relay.
EURASIP Journal on Advances in Signal Processing 5 distributed (i.i.d.) random variables, each having a uniform each having a uniform distribution over the interval [0, 2π ]. distribution over the interval [0, 2π ]. In (3), Furthermore, in (6), (ln) (l) (n) (ln) gS-R ) = a(m) bRk) cS-R ) , (mk ( (mk gR-D = aR bD cR-D , (7a) (4a) S (l) (m) a(l) = e j (π/λ)δR cos(φR-D −βR ) , a(m) = e j (π/λ)δS cos(φS −βS ) (7b) , (4b) R S (n) (k ) bD ) = e j (π/λ)δD cos(φD (n −βD ) bRk) = e j (π/λ)δR cos(φS-R −βR ) , ( , (7c) (4c) (n) (l) (m) (k ) (ln) cR-D = e j (2π/λ){RD cos(φD −γD )−RR cos(φR-D −γD )} (mk) cS-R = e j (2π/λ){RS cos(φS −γS )−RR cos(φS-R −γS )} , (7d) , (4d) 2π 2π θR-D = − θS-R = − (RR + DR-D + RD ), (7e) (RS + DS-R + RR ), (4e) λ λ (l) (l) fS(m) = fSmax cos φSm) − αS , ( fR-D = fRmax cos φR-D − αR , (7f) (4f) fDn) = fDmax cos φD ) − αD , ( (n (k ) (k ) fS-R = fRmax cos φS-R − αR , (7g) (4g) where fDmax = vD /λ is the maximum Doppler frequency where fSmax = vS /λ ( fRmax = vR /λ) is the maximum Doppler caused by the movement of the destination mobile station. frequency caused by the motion of the source mobile station The symbol γD in (7d) refers to the position of the mobile (mobile relay) and λ denotes the carrier’s wavelength. The relay with respect to the destination mobile station (see knowledge of the position of the mobile relay with respect to (l) the source mobile station is incorporated in (4d). It should Figure 1). It is worth highlighting that the AOD φR-D and the ( n) be pointed out here that in the reference model, the AOD AOA φD are independent random variables. Also keep in ( m) (k ) mind that φR-D and φD ) are determined by the distribution (l) (n φS , and the AOA φS-R are independent random variables determined by the distribution of the local scatterers around of the local scatterers around the mobile relay and the the source mobile station and the mobile relay, respectively. destination mobile station, respectively. ( m) ∗ ( m) (k ) (iq) Replacing aS and bR by their complex conjugates aS One can show that the diﬀuse components hR-D (t ) (i, q = (k )∗ and bR in (4b) and (4c), respectively, we can obtain the 1, 2) of the remaining transmission links from the mobile diﬀuse component h(22) (t ) of the A(2) − A(2) transmission (q ) relay antenna element AR to the destination mobile station S-R S R link [30]. The diﬀuse components h(12) (t ) and h(21) (t ) can ( i) antenna element AD can similarly be realized as described S-R S-R ( m) ∗ ( m) (k ) be realized likewise by substituting aS → aS and bR → (q ) for the AS − A(i) transmission link. R bRk)∗ , respectively, in (3) [30]. ( Finally, substituting (1) in (5) allows us to identify the In the same way, it is evident from Figure 2 that the overall fading channel between the source mobile station and input-output relationship of the R-D MIMO subsystem can the destination mobile station as be written as R(t ) = HR-D (t )HS-R (t )S(t ) + HR-D (t )NR (t ) + ND (t ) R(t ) = HR-D (t )X(t ) + ND (t ), (5) (8) = HS-R-D (t )S(t ) + NR-D (t ), T where R(t ) = [R(1) (t ) R(2) (t )] is a 2 × 1 received where NR-D (t ) = HR-D (t )NR (t ) + ND (t ) is the total noise of signal vector at the destination mobile station, the system. The symbol HS-R-D (t ) denotes the overall channel T X(t ) = [X (1) (t ) X (2) (t )] is a 2 × 1 signal vector transmitted matrix, which is deﬁned as follows: by the mobile relay, HR-D (t ) is a 2 × 2 R-D fading channel T HS-R-D (t ) = HR-D (t )HS-R (t ) (1) (2) matrix, and ND (t ) = [ND (t ) ND (t )] is a 2 × 1 AWGN ⎛ ⎞⎛ ⎞ vector. By referring to the previous discussion on the h(11) (t ) h(12) (t ) h(11) (t ) h(12) (t ) elements of the matrix HS-R (t ), one can easily show that the R-D R-D S-R S-R =⎝ ⎠⎝ ⎠ diﬀuse component h(11) (t ) of the A(1) − A(1) transmission (21) (22) (21) (22) hR-D (t ) hR-D (t ) hS-R (t ) hS-R (t ) (9) R-D R D link can be expressed as ⎛ ⎞ (11) (12) hS-R-D (t ) hS-R-D (t ) L N =⎝ ⎠. 1 (l) (n) (ln) h(11) (t ) (ln) = lim √ gR-D e j [2π ( fR-D + fD )t+(θR-D +θR-D )] , h(21) (t ) h(22) (t ) R-D L → ∞ LN l=1 n=1 S-R-D S-R-D N →∞ (6) It is noteworthy that the overall channel matrix HS-R-D (t ) √ (ln) where the term 1/ LN and the symbol θR-D represent the describes completely the reference model of the proposed joint gains and joint phases, respectively, introduced by the geometrical three-ring MIMO M2M fading channel. Here, local scatterers S(l) and S(n) . Like the joint phases, θS-R ) , (mk (iq) each element hS-R-D (t ) (i, q = 1, 2) of the channel matrix R D (ln) deﬁnes the diﬀuse component of the overall MIMO M2M θR-D are assumed to be i.i.d. random variables as well,
6 EURASIP Journal on Advances in Signal Processing (k ) ( n) {φS-R , φD }. Substituting (10) and (11) in (12), the 4D space- fading channel, describing the transmission link from the (q ) time CCF can be expressed as source mobile station antenna element AS to the destination ( i) mobile station antenna element AD via the mobile relay ρ11,22 (δS , δR , δD , τ ) antenna elements. Expanding (9) allows us to explicitly write the diﬀuse component h(11) (t ) of the transmission link S-R-D K L M N Ê 4 2 from the ﬁrst antenna element at the source mobile station, (m)2 (n)2 (k ) (l) = lim E aS bD bR aR A(1) , to the the ﬁrst antenna element at the destination K → ∞ KLMN k=1 l=1 m=1 n=1 S M→∞ L→∞ mobile station, A(1) , as follows: N →∞ D (m) + fS-R + fR-D + fDn) )τ (k ) (l) ( ×e− j 2π ( fS . h(11) (t ) S-R-D (13) (11) (11) (12) (21) = hR-D (t )hS-R (t ) + hR-D (t )hS-R (t ) It is important to highlight, here, the functions of random K L M N Ê 2 (m) (n) (mk) (ln) (k ) (l) = lim √ variables to which the expectation is applied. We can see abc c bR aR KLMN k=1 l=1 m=1 n=1 S D S-R R-D K →∞ that {a(m) , fS(m) } and {a(l) , fR-D } are functions of the AODs (l) L→∞ S R M→∞ ( m) (l) (k ) (k ) ( n) ( n) N →∞ φS and φR-D , respectively. While {bR , fS-R } and {bD , fD } (k ) ( n) are functions of the AOAs φS-R and φD , respectively [30]. (m) (k ) (l) (n) (mk ) (ln) × e j [2π ( fS + fS-R + fR-D + fD )t +(θS-R +θR-D +θS-R +θR-D )] , If the number of local scatterers approaches inﬁnity, that is, (10) K , L, M , N → ∞, then the discrete random variables φSm) , ( (k ) (l) ( n) Ê φS-R , φR-D , and φD become continuous random variables where {bRk) a(l) } denotes the real part of a complex number, ( φS , φS-R , φR-D , and φD , each of which is characterized R Ê (k )∗ (l)∗ (k ) (l) (k ) (l) that is, 2 {bR aR } = bR aR + bR aR . Note that by a certain distribution, denoted by pφS (φS ), pφS-R (φS-R ), the phases θS-R and θR-D in (10) are constant quantities, pφR-D (φR-D ), and pφD (φD ), respectively, [30]. The inﬁnitesimal which can be set to zero without loss of generality since power of the diﬀuse components corresponding to the the statistical properties of the reference model are not diﬀerential angles dφS , dφS-R , dφR-D , and dφD is proportional inﬂuenced by a constant phase shift. Similarly, the diﬀuse to pφS (φS ) pφS-R (φS-R ) pφR-D (φR-D ) pφD (φD )dφS dφS-R dφR-D dφD . (22) (2) (2) component hS-R-D (t ) of the AS − AD transmission link can This implies that when the number of local scatterers approaches inﬁnity, that is, K , L, M , N → ∞, the inﬁnites- be expressed as imal power of all diﬀuse components becomes equal to 1/ (KLMN ), that is, h(22) (t ) S-R-D (21) (12) (22) (22) 1 = hR-D (t )hS-R (t ) + hR-D (t )hS-R (t ) = pφS φS pφS-R φS-R pφR-D φR-D KLMN (14) K L M N Ê 2 (m)∗ (n)∗ (mk) (ln) (k ) (l) = lim √ × pφD φD dφS dφS-R dφR-D dφD . aS bD cS-R cR-D bR aR K → ∞ KLMN m=1 n=1 k=1 l=1 L→∞ M →∞ N →∞ Thus, we can write the 4D space-time CCF ρ11,22 (δS , δR , δD , τ ) of the reference model given in (13) as (m) (k ) (l) (n) (mk ) (ln) − j [2π ( fS − fS-R + fR-D − fD )t −(θS-R +θR-D +θS-R +θR-D )] ×e . (11) ρ11,22 (δS , δR , δD , τ ) = ρS (δS , τ ) · ρR (δR , τ ) · ρD (δD , τ ), (15) The equations (10) and (11) will be used in the next subsection to calculate the 4D space-time CCF. where 3.2. Correlation Properties of the Reference Model. By deﬁni- π a2 δS , φS e− j 2π fS (φS )τ pφS φS dφS , ρ S (δS , τ ) = (16) tion, the 4D space-time CCF between the transmission links S −π (1) (1) (2) (2) AS − AD and AS − AD is equivalent to the correlation π between the diﬀuse components h(11) (t ) and h(22) (t ), that e− j 2π [ fS-R (φS-R )+ fR-D (φR-D )]τ ρR (δR , τ ) = 4 S-R-D S-R-D is, [26] −π Ê bR δR, φS-R aR δR, φR-D (17) 2 × ∗ ρ11,22 (δS , δR , δD , τ ) := E h(11) (t )h(22) (t + τ ) . (12) S-R-D S-R-D × pφS-R φS-R pφR-D φR-D dφS-R dφR-D , It should be noticed here that the expectation operator is π applied to all random variables, that is, the random phase bD δD , φD e− j 2π fD (φD )τ pφD φD dφD 2 ρ D (δD , τ ) = (18) (mk) (ln) ( m) ( l ) shifts {θS-R , θR-D }, the AODs {φS , φR-D }, and the AOAs −π
EURASIP Journal on Advances in Signal Processing 7 are the CFs at the source mobile station, mobile relay, and the respectively, as destination mobile station. Here, π e− j 2π fS (φS )τ pφS φS dφS , ρS (0, τ ) = (23) j (π/λ)δS cos(φS −βS ) aS δS , φS = e , (19a) −π π bR δR , φS-R = e j (π/λ)δR cos(φS-R −βR ) , e− j 2π [ fS-R (φS-R )+ fR-D (φR-D )]τ (19b) ρR (0, τ ) = 4 −π (24) aR δR , φS-R = e j (π/λ)δR cos(φR-D −βR ) , (19c) × pφS-R φS-R pφR-D φR-D dφS-R dφR-D , bD δD , φD = e j (π/λ)δD cos(φD −βD ) , (19d) π e− j 2π fD (φD )τ pφD φD dφD . ρD (0, τ ) = (25) fS φS = fSmax cos φS − αS , −π (19e) fS-R φS-R = fRmax cos φS-R − αR , (19f) The reference model developed in Section 3 is a theo- retical model, which is based on the assumption that the fR-D φR-D = fRmax cos φR-D − αR , (19g) number of local scatterers (K , L, M , N ) around the source mobile station, the mobile relay, and the destination mobile fD φD = fDmax cos φD − αD . (19h) station is inﬁnite. The assumption of an inﬁnite number of local scatterers prevents the realization of the reference In this article, we refer to the CF at the source mobile station model. However, a realizable stochastic simulation model ρS (δS , τ ) as the source CF. Likewise, the CF at the mobile relay can be derived from the reference model by: (i) using only (destination mobile station) ρS (δR , τ ) (ρS (δD , τ )) is termed as a limited number of local scatterers K = L, M , and N is the relay CF (destination CF). Equation (15) illustrates that ﬁnite, (ii) setting the constant phase shifts θS-R and θR-D to the 4D space-time CCF ρ11,22 (δS , δR , δD , τ ) of the reference ( m) zero in (10), and (iii) considering the discrete AOD φS model can be expressed as the product of the source CF and φR-D , as well as the AOA φS-R and φD ) are constant (l) (k ) (n ρS (δS , τ ), the relay CF ρR (δR , τ ), and the destination CF quantities [30]. Thus, using (10), the diﬀuse component ρD (δD , τ ). Besides, from (16) and (18), it turns out that the h(11) (t )(throughout this paper, the caret is used for the source mobile station and the destination mobile station are S-R-D (1) (1) interchangeable. stochastic simulation model) of the AS − AD transmission The 3D spatial CCF ρ(δS , δR , δD ), deﬁned as ρ(δS , δR , link of the stochastic simulation model can be expressed as ∗ δD ) = E{h(11) (t )h(22) (t )}, is equal to the 4D space-time S-R-D S-R-D CCF ρ11,22 (δS , δR , δD , τ ) at τ = 0, that is, h(11) (t ) S-R-D K L M N Ê ρ(δS , δR , δD ) = ρ11,22 (δS , δR , δD , 0) 2 a(m) b(n) c(mk) c(ln) bRk) a(l) ( =√ (20) KLMN k=1 l=1 m=1 n=1 S D S-R R-D R = ρS (δS , 0) · ρR (δR , 0) · ρD (δD , 0). (m) + fS-R + fR-D + fDn) }t −θS-R ) −θR-D ] (k ) (l) ( (mk (ln) × e j [2π { fS . The temporal ACF rh(iq) (τ ) of the diﬀuse component (26) S-R-D (iq) hS-R-D (t ) of the transmission link from the source mobile (q ) station antenna element AS to the destination mobile (22) (2) In the same way, the diﬀuse component hS-R-D (t ) of the AS − station antenna element A(i) , for all i,q ∈ {1, 2} can be given A(2) transmission link in the stochastic simulation model can D D as be expressed as (iq)∗ (iq) rh(iq) (τ ) := E hS-R-D (t )hS-R-D (t + τ ) . (21) h(22) (t ) S-R-D S-R-D K L M N Ê 2 It is not diﬃcult to show that the temporal ACF rh(iq) (τ ), (m)∗ (n)∗ (mk) (ln) (k ) (l) =√ a bD cS-R cR-D bR aR KLMN k=1 l=1 m=1 n=1 S S-R-D for all i,q ∈ {1, 2} is equal to the 4D space-time CCF ρ11,22 (δS , δR , δD , τ ) of the reference model at δS = δR = δD = (m) (k ) (l) (n) (mk ) (ln) × e j [2π { fS + fS-R + fR-D + fD }t −θS-R −θR-D ] . 0, that is, (27) rh(iq) (τ ) = ρ11,22 (0, 0, 0, τ ) S-R-D The 4D space-time CCF of the diﬀuse components h(11) (t ) (22) S-R-D = ρS (0, τ ) · ρR (0, τ ) · ρD (0, τ ). and h(22) (t ) of the stochastic simulation model is deﬁned by S-R-D Substituting δS = 0, δR = 0, and δD = 0 in (16), (17), ∗ ρ11,22 (δS , δR , δD , τ ) := E h(11) (t )h(22) (t + τ ) , (28) S-R-D S-R-D and (18), respectively, gives ρS (0, τ ), ρR (0, τ ), and ρD (0, τ ),
8 EURASIP Journal on Advances in Signal Processing 4. Scattering Scenarios where the expectation operator now only applies on the random phases θS-R ) and θR-D . The substitution of (26) and (mk (ln) This section studies the correlation properties of the ref- (27) in (28) results in the following closed-form expression: erence model under nonisotropic scattering conditions. In order to characterize nonisotropic scattering around the ρ11,22 (δS , δR , δD , τ ) source mobile station (destination mobile station), we have utilized the von Mises distribution for the AOD φS (AOA φD ), K L M N Ê 4 2 that is, a(m)2 bD )2 (n bRk) a(l) ( = KLMN k=1 l=1 m=1 n=1 S R (29) 1 (0) eκS cos(φS −φS ) , pφS φS = φS ∈ [0, 2π ), (m) + fS-R + fR-D + fDn) }τ (k ) (l) ( × e − j 2π { f S 2π I0 (κS ) (36) = ρS (δS , τ ) · ρR (δR , τ ) · ρD (δD , τ ), where I0 (·) is the modiﬁed Bessel function of the ﬁrst kind where (0) (0) of order zero, the parameter φS (φD ) ∈ [0, 2π ) is the mean AOD (AOA), and the parameter κS (κD ) ≥ 0 controls the M 1 (m) (m)2 a S ( δ S ) e − j 2π f S τ , ρ S (δS , τ ) = (30) angular spread of φS (φD ). Similarly, nonisotropic scattering M m=1 around the mobile relay can be deﬁned by the von Mises distribution of the AOA φS-R along with the von Mises K L Ê 4 2 distribution of the AOD φR-D over [0, 2π ). Hence, the (k ) (l) bRk) a(l) ( e− j 2π [ fS-R + fR-D ]τ , ρ R (δR , τ ) = R KL k=1 l=1 distributions pφS-R (φS-R ) and pφR-D (φR-D ) can be obtained by replacing the index S by S-R and R-D in (36), respectively. (31) The reason for using the von Mises distribution is its ﬂexibility to closely approximate the Gaussian distribution N 1 (n) (n)2 bD (δD )e− j 2π fD τ ρ D (δD , τ ) = and the cardioid distribution as well as to include the (32) N n=1 uniform distribution as a special case [35]. Moreover, Abdi et al. [36] made a proposition of employing the are called the source CF, the relay CF, and the destination CF von Mises distribution to model AOA statistics of mobile of the simulation model. radio fading channels. They supported their proposal by The 3D spatial CCF ρ(δS , δR , δD ) of the simulation model, matching the von Mises distribution to the measured data. deﬁned as Substituting (36) in (16) and using [37, Equations (3.338– ∗ 4)] results in the following closed-form expression for ρ(δS , δR , δD ) = E h(11) (t )h(22) (t ) (33) S-R-D S-R-D the source CF: is equal to the 4D space-time CCF ρ11,22 (δS , δR , δD , τ ) at τ = 0, that is, ρ S (δS , τ ) ρ(δS , δR , δD ) = ρ11,22 (δS , δR , δD , 0) 1 = I0 (34) I0 (κS ) = ρS (δS , 0) · ρR (δR , 0) · ρD (δD , 0). 2 δS δS 2 κ2 − 4π 2 In the stochastic simulation model, the temporal ACF × + fSmax τ −2 fS τ cos αS−βS S λ max λ (iq) (q ) ( i) rh(iq) (τ ) of the diﬀuse component hS-R-D (t ) of the AS − AD S-R-D transmission link can be derived as follows: 1/ 2 δS (0) (0) cos βS − φS − fSmax τ cos αS − φS + j 4πκS . ∗ λ (iq) (iq) rh(iq) (τ ) := E hS-R-D (t )hS-R-D (t + τ ) (37) S-R-D K L M N 1 (m) (k ) (l) (n) e− j 2π { fS + fS-R + fR-D + fD }τ = The destination CF ρD (δD , τ ) can easily be realized by KLMN k=1 l=1 m=1 n=1 replacing the index S by D in (37). Likewise, substituting the von Mises distribution for = ρS (0, τ ) · ρR (0, τ ) · ρD (0, τ ) ∀i, q ∈ {1, 2}. the AOA φS-R and the AOD φR-D in (17) and solving the (35) integrals using [37, Equations (3.338-4)], provides us with the closed-form solution for the relay CF ρR (δR , τ ) as given in From (35), it can be seen that the temporal ACF rh(iq) (τ ) (38). S-R-D is equal to the 4D space-time CCF ρ11,22 (δS , δR , δD , τ ) of the Substituting (37), (38), and the closed-form expression simulation model at δS = δR = δD = 0, that is, rh(iq) (τ ) = of ρD (δD , τ ) in (15) results in the 4D space-time CCF S-R-D ρ11,22 (0, 0, 0, τ ). ρ11,22 (δS , δR , δD , τ ) of the reference model in a closed form.
EURASIP Journal on Advances in Signal Processing 9 1 ρ R (δR , τ ) = (I0 (κS-R )I0 (κR-D )) 2 δR δR 2 κ2 − 4π 2 × I0 −2 fR τ cos αR − βR + fRmax τ S-R λ max λ 1/ 2 δR (0) (0) cos βR − φS-R − fRmax τ cos αR − φS-R + j 4πκS-R λ 2 δR δR 2 κ2 − 4π 2 × I0 −2 fR τ cos αR − βR + fRmax τ R-D λ max λ 1/ 2 δR (0) (0) cos βR − φR-D − fRmax τ cos αR − φR-D + j 4πκR-D λ 1/ 2 (0) κ2 − 4π 2 fRmax τ 2 − j 4πκS-R fRmax τ cos αR − φS-R 2 (38) + I0 S-R 1/ 2 (0) κ2 − 4π 2 fRmax τ 2 − j 4πκR-D fRmax τ cos αR − φR-D 2 × I0 R-D 2 δR δR 2 κ2 − 4π 2 fR τ cos αR − βR + I0 + fRmax τ +2 S-R λ max λ 1/ 2 δR (0) (0) − j 4πκS-R cos βR − φS-R + fRmax τ cos αR − φS-R λ 2 δR δR 2 κ2 − 4π 2 × Io fR τ cos αR − βR + fRmax τ +2 R-D λ max λ 1/ 2 δR (0) (0) − j 4πκR-D cos βR − φR-D + fRmax τ cos αR − φR-D . λ 5. Numerical Results Note that the von Mises distribution reduces to the uniform distribution for κS = κS-R = κR-D = κD = The purpose of this section is to illustrate the important 0. This implies that setting the respective κ’s to zero theoretical results found for the CCFs of the reference reduces (37) and (38) to the CFs (i.e., [38, equation model and the stochastic simulation model by evaluating (27)] and [38, Equation (28)], respectively), derived for the expressions in (16), (18), (30), and (32). Here, we focus isotropic scattering conditions. From the 4D space-time on discussing numerical results for the source CFs and the CCF ρ11,22 (δS , δR , δD , τ ) of the reference model derived relay CFs. The results for the destination CFs can easily be for isotropic scattering conditions, it follows that the obtained from the source CFs just by replacing the index S 3D spatial CCF ρ(δS , δR , δD ) of the reference model is a by D. As a performance criterion, we consider the absolute product of four Bessel functions, that is, ρ(δS , δR , δD ) = error eS (δS , τ ) = |ρS (δS , τ ) − ρS (δS , τ )| as a measure for the ρ11,22 (δS , δR , δD , 0) = J0 (2πδS /λ){2J0 (2πδR /λ)}2 J0 (2πδD /λ). quality of the approximation ρS (δS , τ ) ≈ ρS (δS , τ ). Similarly, In the same way, the temporal ACF rh(iq) (τ ) of the reference the absolute error eR (δR , τ ) = |ρR (δR , τ ) − ρR (δR , τ )| has S-R-D model can be written as rh(iq) (τ ) = ρ11,22 (0, 0, 0, τ ) = been introduced to study the amount of precision of the S-R-D approximation ρR (δR , τ ) ≈ ρR (δR , τ ). The selected values for J0 (2π fSmax τ ){2J0 (2π fRmax τ )}2 J0 (2π fDmax τ ). A product of two the parameters inﬂuencing the CFs are: βS = βR = π/ 2, Bessel functions describes the 2D spatial CCF and the αS = π/ 4, αR = 0, and fSmax = fRmax = 91 Hz. The wavelength temporal ACF of the reference model derived from a λ was set to λ = 0.15 m. geometrical two-ring scattering model [30]. For the 3D spatial CCF and the temporal ACF of the reference model derived from a geometrical three-ring scattering model, 6. The Stochastic Simulation Model a product of four Bessel functions is justiﬁed, since the geometrical three-ring scattering model is a concate- For the stochastic simulation model, an appropriate number nation of two separate geometrical two-ring scattering of discrete scatterers M and K (L) located on the rings models. around the source mobile station and the mobile relay,
10 EURASIP Journal on Advances in Signal Processing ×10−3 1.2 1 Source correlation function, ρS (δS , τ ) Absolute error, 1 0.8 0.8 e S (δ S , τ ) 0.6 0.6 0.4 0.4 0.2 0.2 0 0 0 0 1 1 5 5 2 2 4 4 3 3 3 3 2 2 δS / λ δS / λ 4 4 1 1 τ · f Sma τ · f Sma 5 x x 5 0 0 Figure 3: The source CF ρS (δS , τ ) of the 2 × 2 × 2 MIMO M2M Figure 5: Absolute error eS (δS , τ ) = |ρS (δS , τ ) − ρS (δS , τ )| by using the MMEA with M = 40 (isotropic scattering). reference channel model under isotropic scattering conditions. 4 1 Transmit correlation function, ρR (δR , τ ) function, ρS (δS , τ ) Relay correlation 3 0.8 0.6 ^ 2 0.4 1 0.2 0 0 0 0 1 5 3 2 4 1 2.5 3 3 2 2 δS / λ 4 1.5 2 δR / λ 1 τ· f 1 S max 5 0 0.5 · f Rmax 3 τ 0 Figure 4: The source CF ρS (δS , τ ) of the 2 × 2 × 2 MIMO M2M Figure 6: The relay CF ρR (δR , τ ) of the 2 × 2 × 2 MIMO M2M stochastic channel simulator under isotropic scattering conditions. reference channel model under isotropic scattering conditions. respectively, should be selected. In our simulations, we have 4 function, ρR (δR , τ ) chosen M = 40 and K = L = 23. A good solution Relay correlation 3 to the parameter computation problem in M2M fading ^ channel simulators in case of isotropic scattering is advanced 2 in [30], where the authors have suggested the extended 1 method of exact Doppler spread (EMEDS). Whereas, in case 0 of nonisotropic scattering, a high-performance parameter 0 0.5 1 computation method is the modiﬁed method of equal areas 3 2.5 1.5 2 1.5 2 (MMEA) [39]. Since the MMEA reduces to the EMEDS in 2.5 3 δR / λ 1 0.5 τ · f Rmax 3.5 0 case of isotropic scattering [30], we have used the MMEA for ( m) (l) (k ) computing the AODs φS and φR-D as well as the AOA φS-R . Figure 7: The relay CF ρR (δR , τ ) of the 2 × 2 × 2 MIMO M2M Figures 3–8 bring to light the numerical results associated stochastic channel simulator under isotropic scattering conditions. with isotropic scattering conditions. Figure 3 demonstrates the shape of the source CF ρS (δS , τ ) of the reference model determined by (37) when κS = 0, whereas the simulation in the range [0, (K = L)/ 8], the absolute error eR (δR , τ ) is model’s source CF ρS (δS , τ ) is displayed in Figure 4. The almost zero. absolute error eS (δS , τ ), presented in Figure 5, shows the Figures 9–14 elucidate the results in case of nonisotropic quality of the approximation ρS (δS , τ ) ≈ ρS (δS , τ ). The shape scattering. As mentioned in Section 4, the von Mises dis- of the relay CF ρR (δR , τ ) of the reference model given by (38) tribution has been employed to characterize nonisotropic when κR = 0 and the simulation model’s relay CF ρR (δR , τ ) scattering around the source mobile station (destination are exhibited in Figures 6 and 7, respectively. A careful mobile station) and the mobile relay. In our simulations, study of the absolute error eR (δR , τ ) in Figure 8 reveals the the parameters of the von Mises distribution were set as ranges of δR /λ and τ · fRmax with an excellent approximation (0) (0) (0) φS = φS-R = φR-D = 60◦ and κS = κS-R = κR-D = 40. ρR (δR , τ ) ≈ ρR (δR , τ ). When δR /λ is conﬁned in the range [0, (K = L)/ 8], then the approximation ρR (δR , τ ) ≈ ρR (δR , τ ) Figure 9 has been included here to get a clear picture of the is very accurate. On the other hand, for δR = 0 with τ · fRmax absolute value of the reference model’s source CF |ρS (δS , τ )|
EURASIP Journal on Advances in Signal Processing 11 0.6 0.04 0.5 Absolute error, Absolute error, 0.03 e R (δ R , τ ) 0.4 eS (δS , τ ) 0.3 0.02 0.2 0.01 0.1 0 0 0 0 0.5 1 5 3 1 2 4 2.5 1.5 2 3 3 1.5 2 2 δR / λ δS / λ 1 4 2.5 1 0.5 τ · f Sma f Rmax x 5 τ· 0 3 0 Figure 11: Absolute error eS (δS , τ ) = |ρS (δS , τ ) − ρS (δS , τ )| by Figure 8: Absolute error eR (δR , τ ) = |ρR (δR , τ ) − ρR (δR , τ )| by using using the MMEA with M = 40 (nonisotropic scattering, von Mises the MMEA with K = L = 23 (isotropic scattering). density with φS = 60◦ and κS = 40). (0) 1 Transmit correlation function, |ρS (δS , τ )| 4 function, |ρR (δR , τ )| 0.8 Relay correlation 0.6 3 0.4 2 0.2 1 0 0 0 1 5 0 2 4 3 0.5 3 3 1 2 δS / λ 2.5 4 1 1.5 2 τ · f Sma x 5 0 1.5 2 δR / λ 1 2.5 0.5 f Rmax τ· 3 0 Figure 9: Absolute value of the source CF |ρS (δS , τ )| of the 2 × 2 × 2 MIMO M2M reference channel model under nonisotropic Figure 12: Absolute value of the relay CF |ρR (δR , τ )| of the 2 × scattering conditions (von Mises density with φS = 60◦ and κS = (0) 2 × 2 MIMO M2M reference channel model under nonisotropic scattering conditions (von Mises density with φS-R = φR-D = 60◦ (0) (0) 40). and κS-R = κR-D = 40). 1 Transmit correlation function, |ρS (δS , τ )| for computing the simulation model parameters when the 0.8 AODs are non-uniformly distributed on the rings around 0.6 the source mobile station [30]. The successful application of ^ 0.4 the LPNM for minimizing the error function of the one-ring 0.2 model parameters and the two-ring model parameters can be found in [19] and [20], respectively. It is, therefore, believed 0 0 1 that the LPNM is equally beneﬁcial for the computation 5 2 4 3 3 of the three-ring model parameters under nonisotropic 2 δS / λ 4 1 τ · f Sma x 5 0 scattering conditions. The absolute value of the reference model’s relay CF |ρR (δR , τ )| (see (38)) and the absolute Figure 10: Absolute value of the source CF |ρS (δS , τ )| of the 2 × 2 × 2 value of the simulation model’s source CF |ρR (δR , τ )| (see MIMO M2M stochastic channel simulator designed by applying the (31)) are shown in Figures 12 and 13, respectively. The MMEA with M = 40 (nonisotropic scattering, von Mises density measure of the quality of the approximation ρR (δR , τ ) ≈ (0) with φS = 60◦ and κS = 40). ρR (δR , τ ), that is, the absolute error eR (δR , τ ) is illustrated in Figure 14. It can be seen in Figure 14 that the maximum value of eR (δR , τ ) is less than 10−1 . The same arguments given for minimizing eS (δS , τ ) by using the LPNM are valid for given in (37), whereas Figure 10 provides information about minimizing eR (δR , τ ). the absolute value of the simulation model’s source CF |ρS (δS , τ )|. The absolute error eS (δS , τ ) in Figure 11 shows that the approximation ρS (δS , τ ) ≈ ρS (δS , τ ) holds for 7. Conclusion nonisotropic scattering as well. The error function eS (δS , τ ) shows a ripple eﬀect, where the maximum value of this error In this article, we have derived a reference model and function is in the orders of 3 · 10−2 . It has been recommended a stochastic simulation model for narrowband MIMO in the literature to utilize the Lp-norm method (LPNM) [40] M2M fading channel for relay-based cooperative networks.
12 EURASIP Journal on Advances in Signal Processing Acknowledgment 4 function, |ρR (δR , τ )| The material in this paper was presented in part at the IEEE Relay correlation 69th Vehicular Technology Conference, VTC-Spring 2009, 3 Barcelona, Spain, April 2009. ^ 2 1 References 0 0 0.5 3 1 [1] S. Barbarossa and G. Scutari, “Cooperative diversity through 2.5 2 1.5 1.5 2 virtual arrays in multihop networks,” in Proceedings of the δR / λ 1 2.5 0.5 f Rmax τ· 3 0 IEEE International Conference on Accoustics, Speech, and Signal Processing, pp. 209–212, Hong Kong, April 2003. [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation Figure 13: Absolute value of the relay CF |ρR (δR , τ )| of the 2 × 2 × diversity—part I: system description,” IEEE Transactions on 2 MIMO M2M stochastic channel simulator designed by applying Communications, vol. 51, no. 11, pp. 1927–1938, 2003. the MMEA with K = L = 23 (nonisotropic scattering, von Mises [3] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation (0) (0) density with φS-R = φR-D = 60◦ and κS-R = κR-D = 40). diversity—part II: implementation aspects and performance analysis,” IEEE Transactions on Communications, vol. 51, no. 11, pp. 1939–1948, 2003. [4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: eﬃcient protocols and outage behavior,” IEEE Transactions on Information Theory, vol. 50, no. 12, pp. 3062–3080, 2004. 0.2 [5] A. S. Akki and F. Haber, “A statistical model of mobile-to- Absolute error, 0.15 mobile land communication channel,” IEEE Transactions on eR (δR , τ ) Vehicular Technology, vol. VT-35, no. 1, pp. 2–7, 1986. 0.1 [6] A. S. Akki, “Statistical properties of mobile-to-mobile land 0.05 communication channels,” IEEE Transactions on Vehicular Technology, vol. 43, no. 4, pp. 826–831, 1994. 0 3 [7] B. Talha and M. P¨ tzold, “On the statistical properties of a 2.5 3 2 2.5 2 1.5 double Rice channels,” in Proceedings of the 10th International 1.5 1 δR / λ 1 0.5 Symposium on Wireless Personal Multimedia Communications 0.5 τ · f Rmax 0 0 (WPMC ’07), pp. 517–522, Jaipur, India, December 2007. [8] B. Talha and M. P¨ tzold, “Statistical modeling and analysis a Figure 14: Absolute error eR (δR , τ ) = |ρR (δR , τ ) − ρR (δR , τ )| by of mobile-to-mobile fading channels in cooperative networks using the MMEA with K = L = 23 (nonisotropic scattering, von under line-of-sight conditions,” Wireless Personal Communi- Mises density with φS-R = φR-D = 60◦ and κS-R = κR-D = 40). (0) (0) cations, vol. 54, no. 1, pp. 3–19, 2010. [9] B. Talha and M. P¨ tzold, “Mobile-to-mobile fading chan- a nels in amplify-and-forward relay systems under line-of- sight conditions: statistical modeling and analysis,” Annals of Telecommunications, vol. 65, no. 7-8, pp. 391–410, 2010. The starting point for deriving the reference model was [10] V. Erceg, S. J. Fortune, J. Ling, A. J. Rustako Jr., and R. A. the geometrical three-ring scattering model, where it is Valenzuela, “Comparisons of a computer-based propagation assumed that the local scatterers are located on rings prediction tool with experimental data collected in urban around the source mobile station, the mobile relay, and microcellular environments,” IEEE Journal on Selected Areas in Communications, vol. 15, no. 4, pp. 677–684, 1997. the destination mobile station. Furthermore, the suggested [11] C. S. Patel, G. L. St¨ ber, and T. G. Pratt, “Statistical properties u three-ring scattering model came out to be a concatenation of amplify and forward relay fading channels,” IEEE Transac- of two separate two-ring scattering models. General ana- tions on Vehicular Technology, vol. 55, no. 1, pp. 1–9, 2006. lytical formulas along with exact closed-form expressions [12] I. Z. Kov´ cs, P. C. F. Eggers, K. Olesen, and L. G. Petersen, a for the source (destination) CF and the relay CF speciﬁc “Investigations of outdoor-to-indoor mobile-to-mobile radio to nonisotropic scattering have been presented. The results communication channels,” in Proceedings of the 56th Vehicular show that various CCFs describing the simulation model Technology Conference (VTC ’02), pp. 430–434, Vancouver, closely approximate the corresponding CCFs of the reference Canada, September 2002. model. The developed channel simulator is useful for [13] J. B. Andersen, “Statistical distributions in mobile communi- analyzing the dynamic behavior of the MIMO channel cations using multiple scattering,” in Proceedings of the 27th capacity of relay-based M2M communication systems. In URSI General Assembly, Maastricht, The Netherlands, August addition, the proposed geometrical three-ring scattering 2002. model for narrowband MIMO M2M fading channels is the [14] J. Salo, H. M. El-Sallabi, and P. Vainikainen, “Statistical groundwork towards the development and analysis of new analysis of the multiple scattering radio channel,” IEEE channels models for wideband MIMO M2M fading channels Transactions on Antennas and Propagation, vol. 54, no. 11, pp. in cooperative networks. 3114–3124, 2006.
EURASIP Journal on Advances in Signal Processing 13 [15] G. J. Foschini and M. J. Gans, “On limits of wireless com- [30] M. P¨ tzold, B. O. Hogstad, and N. Youssef, “Modeling, a munications in a fading environment when using multiple analysis, and simulation of MIMO mobile-to-mobile fading antennas,” Wireless Personal Communications, vol. 6, no. 3, pp. channels,” IEEE Transactions on Wireless Communications, vol. 311–335, 1998. 7, no. 2, pp. 510–520, 2008. [16] E. Telatar, “Capacity of multi-antenna Gaussian channels,” [31] A. G. Zaji´ and G. L. St¨ ber, “Space-time correlated mobile- c u European Transactions on Telecommunications, vol. 10, no. 6, to-mobile channels: modelling and simulation,” IEEE Trans- pp. 585–595, 1999. actions on Vehicular Technology, vol. 57, no. 2, pp. 715–726, 2008. [17] C. S. Patel, G. L. St¨ ber, and T. G. Pratt, “Comparative analysis u of statistical models for the simulation of Rayleigh faded [32] A. Chelli and M. P¨ tzold, “A MIMO mobile-to-mobile a cellular channels,” IEEE Transactions on Communications, vol. channel model derived from a geometric street scattering 53, no. 6, pp. 1017–1026, 2005. model,” in Proceedings of the 4th IEEE International Symposium on Wireless Communication Systems 2007 (ISWCS ’07), pp. [18] D. S. Shiu, G. J. Foschini, M. J. Gans, and J. M. Kahn, “Fading 792–797, Trondheim, Norway, October 2007. correlation and its eﬀect on the capacity of multielement antenna systems,” IEEE Transactions on Communications, vol. [33] A. Chelli and M. P¨ tzold, “A non-stationary MIMO vehicle- a 48, no. 3, pp. 502–513, 2000. to-vehicle channel model based on the geometrical T-junction model,” in Proceedings of the International Conference on [19] M. P¨ tzold and B. O. Hogstad, “A space-time channel a Wireless Communications and Signal Processing (WCSP ’09), simulator for MIMO channels based on the geometrical one- Nanjing, China, November 2009. ring scattering model,” Wireless Communications and Mobile Computing, vol. 4, no. 7, pp. 727–737, 2004. [34] A. G. Zaji´ and G. L. St¨ ber, “Three-dimensional modeling, c u simulation, and capacity analysis of space-time correlated [20] M. P¨ tzold and B. O. Hogstad, “Design and performance a mobile-to-mobile channels,” IEEE Transactions on Vehicular of MIMO channel simulators derived from the two-ring Technology, vol. 57, no. 4, pp. 2042–2054, 2008. scattering model,” in Proceedings of the 14th IST Mobile & Communications Summit (IST ’05), Dresden, Germany, June [35] K. V. Mardia and P. E. Jupp, Directional Statistics, Chichester: 2005, paper no. 121. John Wiley & Sons, 1999. [21] G. J. Byers and F. Takawira, “Spatially and temporally corre- [36] A. Abdi, J. A. Barger, and M. Kaveh, “A parametric model lated MIMO channels: modeling and capacity analysis,” IEEE for the distribution of the angle of arrival and the associated Transactions on Vehicular Technology, vol. 53, no. 3, pp. 634– correlation function and power spectrum at the mobile 643, 2004. station,” IEEE Transactions on Vehicular Technology, vol. 51, no. 3, pp. 425–434, 2002. [22] Z. Tang and A. S. Mohan, “A correlated indoor MIMO channel model,” in Proceedings of the Canadian Conference on Electrical [37] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and and Computer Engineering (CCECE ’03), vol. 3, pp. 1889–1892, Products, Academic Press, New York, NY, USA, 6th edition, Venice, Italy, May 2003. 2000. [23] R. B. Ertel, P. Cardieri, K. W. Sowerby, T. S. Rappaport, and [38] B. Talha and M. P¨ tzold, “A geometrical channel model a J. H. Reed, “Overview of spatial channel models for antenna for MIMO mobile-to-mobile fading channels in cooperative array communication systems,” IEEE Personal Communica- networks,” in Proceedings of the 69th IEEE Vehicular Technology tions, vol. 5, no. 1, pp. 10–22, 1998. Conference (VTC ’09), Barcelona, Spain, April 2009. [24] R. B. Ertel and J. H. Reed, “Angle and time of arrival statistics [39] C. A. Gutierrez-Diaz-de-Leon and M. P¨ tzold, “Sum-of- a for circular and elliptical scattering models,” IEEE Journal on sinusoids-based simulation of ﬂat fading wireless propagation Selected Areas in Communications, vol. 17, no. 11, pp. 1829– channels under non-isotropic scattering conditions,” in Pro- 1840, 1999. ceedings of the 50th Annual IEEE Global Telecommunications Conference (GLOBECOM ’07), pp. 3842–3846, Washington, [25] M. P¨ tzold and N. Youssef, “Modelling and simulation a DC, USA, November 2007. of direction-selective and frequency-selective mobile radio channels,” AEU-International Journal of Electronics and Com- [40] M. P¨ tzold, Mobile Fading Channels, John Wiley & Sons, a munications, vol. 55, no. 6, pp. 433–442, 2001. Chichester, UK, 2002. [26] 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. [27] T. A. Chen, M. P. Fitz, W. K. Kuo, M. D. Zoltowski, and J. H. Grimm, “Space-time model for frequency nonselective Rayleigh fading channels with applications to space-time modems,” IEEE Journal on Selected Areas in Communications, vol. 18, no. 7, pp. 1175–1190, 2000. [28] Y. Ma and M. P¨ tzold, “A wideband one-ring MIMO channel a model under non-isotropic scattering conditions,” in Proceed- ings of the IEEE Vehicular Technology Conference (VTC ’08), pp. 424–429, Marina Bay, Singapore, May 2008. [29] M. P¨ tzold and B. O. Hogstad, “A wideband MIMO channel a model derived from the geometric elliptical scattering model,” Wireless Communications and Mobile Computing, vol. 8, no. 5, pp. 597–605, 2008.