On the Capacity and Simulation of 3-D MIMO Mobile-to-Mobile Relay Fading Channels

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1 200 I 2st International yposiu on Personal Indoor and Mobile adio Counications On the Capacity and iulation of 3- MIMO Mobile-to-Mobile elay Fading Channels anouel T. Michailidis Panagiotis Theofilakos and thanasios G. Kanatas epartent of igital ystes University of Piraeus Piraeus Greece {eichail ptheo kanatas}@unipi.gr bstract This paper proposes a theoretical fraework for the evaluation of the capacity of ultiple-int ultiple-outt (MIMO obile-to-obile (M-to-M fading channels in dual-hop aplify-and-forward (F single-relay wireless counications networks. The evaluation is based on a recently introduced threediensional (3- geoetry-based reference odel for these channels. Fro this odel 3- deterinistic and statistical suof-sinusoids (o based siulation odels are also developed under 3- non-isotropic scattering conditions. The perforance of the siulation odels is verified through siulation results. Keywords-plify-and-forward relay links; channel capacity; MIMO channels; obile-to-obile channels; siulation; 3- scattering I. INTOUCTION Mobile-to-Mobile (M-to-M counications intend to substantially contribute to obile ad hoc networks intelligent transportation systes broadband obile ultiedia services and relay-based cellular networks by expanding the network range and increasing the obility support. owever harsh ultipath fading effects usually degrade the transission link quality. Cobining the features of cooperative diversity via wireless relays with the benefits of ultiple-int ultipleoutt (MIMO technology akes it possible to fulfill the growing deands of enhanced reliability and flexibility with increased channel capacity. owever the analysis design and perforance evaluation of MIMO M-to-M relay-based systes necessitate a thorough investigation of the underlying radio channel. Therefore realistic channel odels are indispensable. In [] [2] two-diensional (2- two-ring odels for MIMO M-to-M channels were proposed while a threediensional (3- two-cylinder odel was presented in [3]. Nevertheless the statistical properties of these channels are quite different fro MIMO M-to-M relay channels where a transceiver unit called a relay ( is interposed between a source ( and a destination (. In this case all the obiles stations are affected by the scatterers in their local vicinity. Based on the two-ring odels a three-ring odel for MIMO M-to-M relay fading channels was proposed in [4] where a separate ring of scatterers around the relay is considered in addition to the rings around the source and the destination. This odel assues that the radio waves travel only in the horizontal plane. owever in densely built-up urban areas the antenna arrays are usually located lower than the surrounding scatterers and the scattered waves propagate by diffraction fro vertical structures i.e. buildings to the street level. Then the waves ay travel in both horizontal and vertical planes i.e. 3- scattering occurs. Thus aziuth and elevation angles for the incidence of the radio waves should be introduced [3]. ecently the authors addressed the design of a 3- reference geoetry-based odel for MIMO M-to-M fading channels in dual-hop aplify-and-forward (F single-relay networks [5]. This odel assues that the scatterers in the vicinity of the source the relay and the destination lie on the surface of three cylinders which reflect the influence of three heterogeneous 3- non-isotropic scattering environents. In this paper using this reference odel the channel capacity of unifor linear antenna arrays (Us is studied and evaluated for different odel paraeters. The reference odel is practically non-realizable since an infinite nuber of local scatterers is assued. By adopting the su-of-sinusoids (o principle [6] due to its reasonably low cotational costs and the explicit inclusion of spatial inforation e.g. the ultipath angles of arrival/departure this paper proposes deterinistic and statistical siulation odels with a finite nuber of scatterers under the fraework of the reference odel. The reainder of the paper is organized as follows. ection II presents the syste odel and defines the channel capacity. ection III details the syste geoetry while ection IV describes the 3- reference odel. eterinistic and statistical siulation odels are developed in ection V. ection VI provides nuerical and siulation results. Finally conclusions are drawn in ection VII. II. YTM MO N CNN CPCITY This paper considers a narrowband MIMO M-to-M relaybased counications syste with and antenna eleents at the source the relay and the destination respectively. ll antennas are oni-directional with low height and are nubered as p p q q and u u respectively. It is assued that the direct link between source and destination is obstructed. Then the entire -- MIMO syste can be separated into the - and - MIMO subsystes. The received signal vector at the relay is given by P y( t ( t s( t + n ( t ( /0/$ I 2370

2 where P is the total source transit power ( t is the channel transfer atrix for the transission link fro the source to the relay ( t s is the signal vector transitted by the source with ss I and ( N n( t NC 0 o I is the zero-ean circularly syetric coplex Gaussian noise vector at the relay where N o is the noise power spectral density (P. The relay noralizes the received signal power to unity which is 2 y ( P + No and retransits it to the destination with ean total power P. The received signal vector at the destination is P y( t ( t y( t + n ( t (2 P N ( + o where ( t is the channel transfer atrix for the transission link fro the relay to the destination. n( t NC ( 0 No I is the noise vector at the destination and N o is the noise P. Using ( and (2 the int-outt relation for the overall -- syste becoes PP y( t ( t s( t + n eff ( t (3 P N ( + o where ( ( ( t t t is the effective channel transfer atrix describing the MIMO relay fading channel and ( t P / ( P + N ( t ( t + ( t n eff o n n is the effective noise of the -- syste. This channel is assued static for two consecutive tieslots i.e. the period needed to coplete the F operation. Conditioning on and the effective noise of the syste is Gaussian distributed with covariance atrix ρ n ( t N ( ( eff o I + t t ( + ρ (4 where I is an identity atrix of size and ρ P / No and ρ P / No denote the ean received signal-to-noise ratio (N per antenna at the relay and destination node respectively. By assuing that the channel is known to the /2 destination ultiplying y on the left by n ( t applying eff the atrix inversion lea to calculate n ( t and using eff known results fro inforation theory we obtain the instantaneous channel capacity (in bits per second per hertz: ( log det + χ ( t C t 2 I ( ( I ( ( ( t G t t t (5 where ρ ρ χ ( + ρ + ( ( ( + (6 ρ G t t t I. (7 ρ The eleents of ( t can be directly generated using the proposed siulation odels in ection V. lternatively since ( t ( t ( t ( t can be indirectly derived using the reference odel in ection IV. In particular ( t can be derived as follows [7] (8 /2 T /2 w where w is a stochastic atrix with independent and identically distributed (i.i.d. zero-ean coplex Gaussian entries and are the source and relay (- subsyste correlation atrices ( /2 denotes the atrix square root operation and ( T denotes the transpose operation. The MIMO channel atrix can be generated by following a siilar procedure to that used to generate. III. T 3- T-CYIN CTTING MO Fig. shows the 3- scattering odel for a 2x2x2 MIMO M-to-M relay fading channel [5]. One observes that the relay is positioned at an angle ω with respect to the source while the location of the relay seen fro the destination is described by the angle ω. The antenna spacing at the source relay and destination is denoted by δ δ and δ respectively while O O and O are the corresponding antenna array centers. Moreover the angles θ θ andθ represent the orientation of each antenna array relative to the x-axis and ψ ψ and ψ describe the elevation angle of the eleents p q and u respectively relative to the x-y plane. In addition the source relay and destination are oving with speeds v v and v respectively while the angles γ γ and γ control the oving directions. Note that denotes the projection of a point onto the x-y plane. It is assued that M scatterers denoted by ( ( 2... M are situated around the source on the surface of a cylinder of radius. iilarly ( k ( l ( k 2... K and ( l 2... scatterers in the ( k ( l vicinity of the relay with K and for k l lie on a surface of a cylinder of radius while N scatterers denoted by ( n ( n 2... N are situated ad the destination on the surface of a cylinder of radius. Only double-bounce non-line-of-sight (No propagation conditions are considered which are doinant in urban 237

3 ( acrocells propagation environents [8]. The angles a and ( β denote the aziuth angle of departure (o and the elevation angle of departure (o respectively of the wave ( transitted fro the source and ipinged on while ( k a ( k and β are the aziuth angle of arrival (o and the elevation angle of arrival (o respectively of the wave scattered fro ( k ( and received at the relay. Besides a l ( l and β denote the o and the o respectively of the wave transitted fro the relay and ipinged on ( l while ( n ( n a and β are the o and the o respectively of the wave scattered fro ( n and received at the destination. ue to the heterogeneity of the scattering environents and the double-bounce scattering the angles of departure are independent fro the angles of arrival while the aziuth and elevation angles are also independent [9]. It is assued that the inequalities ax { } d and { } d hold where d and d denote the distances O to O and O to O respectively. It is also assued that ax { δ δ δ} in { } Finally it is assued that d < 4 /( λ ( ( and d < 4 /( λ ( ( where λ is the carrier wavelength. ence the channel does not experience keyhole behavior [7]. z ( v ( β π γ y x ( k v ( k β π γ ource ψ O ( k a O θ q elay ( l β ( l a estination p u ψ ψ O O p ( a θ v u O π γ θ O q ( l 2- Projection d O ω O d ω O ( n β ( n a Figure. The three-cylinder scattering odel for a 2x2x2 MIMO M-to-M single-relay dual-hop fading channel [5]. ( n IV. T 3- FNC MO The reference odel assues that the nuber of scatterers in the vicinity of the source the relay and the destination is infinite. Then for 2 the received coplex faded envelope of the link fro the source eleent p to the destination eleent u via the relay eleent q is given by [5] K M N h t li C C ( K KMN k l n M N where * * ( B B BB exp( jϕk jϕln jϕ jϕ exp j2 πt f + f + f + f (9 ( exp jπ( 3 2 p δ sinψ sin β ( ( exp jπ ( 3 2 p δ cosψ cos β cos ( θ a ( n exp jπ ( 3 2u δsinψsin β ( n ( n exp jπ ( 3 2u δ cosψ cos β cos ( θ a ( k B exp jπ( 3 2q δ sinψ sin β ( k ( k exp jπ ( 3 2q δ cosψ cos β cos ( θ a ( l B exp jπ( 3 2q δ sinψ sin β ( l ( l exp jπ ( 3 2q δ cosψ cos β cos ( θ a exp { 2 cos cos k C j π a ω ( a ω / λ } ( { } exp 2 cos n cos l C j π a ω a ω / λ ϕ 2 π d + + / λ ( d ϕ 2 π + + / λ ( f f cos a γ cos β ( ( n ( n ( k ( k ( ( l ( l ax f f cos a γ cos β ax f f cos a γ cos β ax f f cos a γ cos β (0 ax and fax v / λ fax v / λ and fax v / λ are the axiu oppler shifts associated with the source destination and relay respectively. The phases ϕ k and ϕ ln are rando variables uniforly distributed in the interval [02π]. The space-tie correlation function (TCF between h ( t and h ( t is given by * ( δ δ δ τ Ε ( ( + τ p u h t hp u t ( 2372

4 where ( * denotes coplex conjugate operation and Ε [] is the statistical expectation operator. ince the nuber of scatterers is infinite the discrete o o o and o can be replaced with continuous rando variables a β a β a β a and β respectively with a joint probability density function (pdf. This joint pdf can be decoposed to a product of eight separate pdfs since the aforeentioned rando variables are utually independent. The von Mises pdf [0] is used to characterize the os and os. Considering the o a this pdf is defined as exp k cos ( a μ f ( a - π a π (2 2π I0 ( k where I0 ( is the zeroth-order odified Bessel function of the first kind μ [ ππ ] is the ean angle at which the scatterers in the vicinity of the source are distributed in the xy - plane and k 0 controls the spread around the ean. The Parson s pdf [] is used to characterize the os and os. Considering the o β this pdf is defined as f ( β π π β π cos β < β 4 β 2 β 2 0 otherwise (3 where β is the axiu value of β usually less than 20 value typical for M-to-M counications [9]. Using (9 (2 and (3 and the results in [2] [4] and [5] ( can be written as p u ( δ δ δ τ ( δ τ ( δ τ ( δ τ where ( δ τ and (4 δ τ denote the source and the destination TCFs respectively q q ( δ τ ( δ τ ( δ τ (5 is the relay TCF of the -- syste and ( δ τ and ( δ τ denote the relay TCF of the - and - systes respectively. The source TCF is derived as [3] [5] 2 2 ( + cos β β π β I0 x y ( δ τ π cos 2 β β 4 0 β I k exp( j2π( p p δ sinψ sin β / λ dβ (6 where x j2π( p p δ cosθ cos ψ / λ j2πτ f cosγ + k cos μ /cos β (7 ax ( y j2π p p δ sinθ cos ψ / λ j2πτ f sinγ + k sin μ /cos β. (8 ax The other TCFs can be siilarly defined fro (6-(8 by replacing the indices. Clearly the TCF in (4 is accurate only over a tie duration that is uch saller than in / v / v / v. { } V. 3- IMUTION MO The reference odel assues an infinite nuber of scatterers and prevents practical realization. owever fro this odel o-based siulation odels with a finite nuber of scatterers can be developed. Using (9 and the results in [3] the following received coplex faded envelope is proposed: ( /2 h t KKM M NN K K M M N N k k l l n n n n * * ( jϕ k k jϕ llnn jϕ jϕ j πt( f fn n fk k fl l C C B B + B B k k l l k k l l k k l l exp exp where K K K M M M N N N ( exp sin sin / jπδ ψ β ( ( exp jπδ cosψ cos β cos ( θ a ( n n exp sin sin / n jπδ ψ β ( n ( n exp jπδ cosψ cos β cos ( θ a ( k B k exp sin sin / k jπδ ψ β ( k ( k exp jπδ cosψ cos β cos ( θ a ( l B l exp sin sin / l jπδ ψ β ( l ( l exp jπδ cosψ cos β cos ( θ a ( C exp{ 2 cos k j π ( a ω ( k cos ( a ω / λ} ( n C n exp{ 2 cos l j π ( a ω ( l cos ( a ω / λ} ( f fax cos a γ cos β ( n ( n ( k ( k ( ( l ( l f f cos a γ cos β n n ax f f cos a γ cos β k k ax l l ax (9 f f cos a γ cos β. (

5 Note that the indices and in (9 and (20 are associated with the ultipath aziuth and elevation angles respectively. Using ( (9 and (20 the TCF between h ( t and ( t h is derived as follows ( δ δ δ τ p u K K M M N N ( KK M M N N k k l l n n * * 2 ( B k k B l l B k k B l l j πτ ( f fn n fk k fl l 2 2 n n + exp (2 To successfully approxiate the statistical properties of the reference odel the ultipath aziuth and elevation angles should be estiated. lthough the p -nor ethod (PNM [2] provides axiu accuracy and versatility it requires the iniization of ultiple error functions in order to obtain proper correlation properties. This iniization ust be siultaneously applied for the aziuth and elevation angles. Thus sipler and less coplex ethods are desirable. First a deterinistic (ergodic statistical odel is developed. This odel cotes fixed values for the aziuth and elevation angles and needs a single siulation trial to obtain the desired statistical properties. Based on [3] we ( generate the o a ( and the o β as follows ( ( a F 0.5 / M (22 ( 2β 2 β arcsin (23 π M for... M and... M. The function F ( denotes the inverse function of the von Mises cuulative distribution function (cdf and can be nuerically evaluated [2]. One can siilarly generate the other aziuth and elevation angles. The coplexity of this odel is reduced by choosing a iniu but also adequate nuber of scatterers. statistical (Monte Carlo odel is also developed. The statistical properties of this odel vary for each siulation trial and converge to the theoretical ones after averaging over a sufficient nuber of siulation trials for an arbitrary finite nuber of scatterers. Based on [3] we generate the o ( ( a and the o β as follows β ( ( 2β 2( + ζ ( a F + δ / M (24 arcsin π M (25 for... M and... M where δ and ζ are independent rando variables uniforly distributed in the interval [0. The other aziuth and elevation angles can be siilarly generated. The statistical odel is ore coplex than the deterinistic odel due to the average procedure and the generation of ultiple rando variables. VI. NUMIC N IMUTION UT This section deonstrates the theoretical derivations described above. Figs. 2-4 study the ergodic capacity C erg [C] of a MIMO M-to-M relay fading channel obtained using the 3- reference odel and (5-(8. Unless indicated otherwise the values of the odel paraeters used are ρ ρ 20 db 2 δ δ δ λ θ θ θ π /2 ψ ψ ψ π / 0 k k k k 5 μ μ μ μ 0 β β β β 20 and fax fax fax 0 z. Fig. 2 shows the C erg as a function of the antenna spacing at the relay for different antenna spacing at the source and destination. One observes that increasing the spacing fro 0.5λ to λ increases the capacity fro 7. to 8.5 bps/z. Nevertheless increasing the spacing beyond λ has a negligible effect on the capacity. Fig. 3 illustrates the shape of C erg for different ean received N at the relay and the destination node. The results reveal that the capacity strongly depends on the N and takes values between 0.8 and 2 bps/z. rgodic capacity (bps/z δ δ 0.5λ δ δ λ δ δ 2λ δ /λ Figure 2. rgodic capacity of a 2x2x2 MIMO M-to-M relay channel for different antenna inter-eleent spacing at the source relay and destination. rgodic capacity (bps/z ρ (db ρ (db Figure 3. rgodic capacity of a 2x2x2 MIMO M-to-M relay channel for different ean received N per antenna at the relay and the destination

6 rgodic capacity (bps/z pace-tie Correlation β β o β β 5 o β β 0 o β β 5 o β β 20 o β (β ( o Figure 4. rgodic capacity of a 2x2x2 MIMO M-to-M relay channel for different axiu os and os and vertically placed antennas eference Model eterinistic Model tatistical Model Noralized Tie elay [f τ] ax Figure 5. eal part of the TCFs of the reference and siulation odels. Fig. 4 justifies the inclusion of the third diension of the odel and shows the C erg as a function of the axiu os and os for ψ ψ ψ π / 2. One observes that increasing β and β fro to 20 decreases the capacity by 0.5 bps/z. owever increasing β and β decreases the capacity loss. Therefore the 2- odel in [4] overestiates the available capacity. Fig. 5 exaines the perforance of the siulation odels and copares the real part of their TCF with the real part of the TCF of the reference odel. The values of the odel paraeters used are 2 δ δ δ λ θ θ θ π /6 ψ ψ ψ π /4 k 8 k k 5 k 2 μ 0 μ μ π /6 μ π /3 β β β β 5 γ 0 γ π /4 and γ π /2. noralized sapling period faxt 0.0 is used where T is the sapling period and f ax f. ax f ax The TCF of the deterinistic odel is obtained for K M N 30 and K M N 4. Besides the TCF of the statistical odel is obtained for K M N 20 K M N 3 and 30 siulation trials. One observes that the TCF of the deterinistic odel closely atches the reference one whereas the TCF of the statistical odel perfectly reproduces the reference one. VII. CONCUION In this paper the capacity of outdoor MIMO M-to-M fading channels in F relay-based networks has been studied under 3- non-isotropic scattering conditions. eterinistic and statistical siulation odels for these channels have been also developed and the inherent advantages of the o principle have been exploited. The results have confired that the siulation odels approxiate the statistical properties of the reference odel with high precision. To further validate the utility of the reference odel future work could be devoted to coparing its statistical properties with easureent data. CKNOWGMNT This work is co-funded by the uropean Union and national resources under the National trategic eference Fraework (NF and the T research project: INTNTION. FNC []. G. Zajic and G.. tüber pace-tie correlated obile-to-obile channels: Modeling and siulation I Trans. on Vehic. Technol. vol. 57 no. 2 pp Mar [2] M. Pätzold B. O. ogstad and N. Youssef Modeling analysis and siulation of MIMO obile-to-obile fading channels I Trans. on Wireless Counications vol.7 no.2 pp Feb [3]. G. Zajic and G.. tüber Three-diensional odeling siulation and capacity analysis of space-tie correlated obile-to-obile channels I Trans. on Vehic. Techn. vol. 57 no. 4 pp Jul [4] B. Talha and M. Pätzold Geoetrical Three-ing-Based Model for MIMO Mobile-to-Mobile Fading Channels in Cooperative Networks UIP Journal on dvances in ignal Processing vol [5] Ε. T. Michailidis P. Theofilakos and. G. Kanatas 3- Model for MIMO Mobile-to-Mobile plify-and-forward elay Fading Channels in Proc. 6th uropean Conference on ntennas and Propagation (ucp 202 Prague Czech eblic Mar [6]. O. ice Matheatical nalysis of ando Noise Bell yste Technical of Journal Vol. 23 pp [7]. Gesbert. Bölcskei.. Gore and. J. Paulraj Outdoor MIMO wireless channels: Models and perforance prediction I Trans. Coun. vol. 50 no. 2 pp ec [8]. G. Zajic G.. tüber T. G. Pratt and. T. Nguyen Wideband MIMO Mobile-to-Mobile Channels: Geoetry-Based tatistical Modeling With xperiental Verification I Trans. on Vehic. Technol vol.58 no.2 pp Feb [9] Y. Yaada Y. bine and N. Nakajia Base station/vehicular antenna design techniques eployed in high capacity land obile counications syste ev. lectr. Coun. ab. vol. 35 no. 2 pp [0]. bdi and M. Kaveh space-tie correlation odel for ultieleent antenna systes in obile fading channels I J. on elec. reas in Counic. vol. 20 no. 3 pp pr [] J.. Parsons and. M.. Turkani Characterisation of obile radio signals:model description in Proc. Inst. lectr. ng. I Coun. peech Vision vol. 38 no. 6 pp ec. 99. [2] K. V. Mardia and. P. Jupp irectional tatistics New York. Wiley