Statistical Modeling of MIMO Mobile-to-Mobile ... - Semantic Scholar

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Statistical Modeling of MIMO Mobile-to-Mobile Underwater Channels Alenka G. Zaji´c, Member, IEEE

Abstract— This paper proposes a geometry-based statistical model for wideband multiple-input multiple-output (MIMO) mobile-to-mobile (M-to-M) shallow water acoustic (SWA) multipath fading channels. From the reference model, the corresponding MIMO time-frequency correlation function, Doppler power spectral density, and delay cross-power spectral density are derived. These statistics are important tools for design and performance analysis of MIMO M-to-M SWA communication and sonar systems. Finally, the derived statistics are compared with the experimentally obtained channel statistics and close agreement is observed. Index Terms— Shallow water acoustic channels, mobile-tomobile underwater channels, multiple-input multiple-output underwater channels, wideband channels, synthetic aperture sonar, underwater sensor networks.

I. I NTRODUCTION There has been a growing interest in designing underwater wireless sensor networks because of their ability to bring computation and sensing into the underwater environment. Underwater networks have many potential applications, including seismic monitoring, scientific exploration of the ocean, tactical surveillance, pollution monitoring, offshore exploration, and support for underwater robots. To make these applications feasible, there is a need for wireless underwater acoustic communications among deployed sensors and autonomous underwater vehicles. Statistical characterization of the acoustic communication channel is necessary in order to assess system performance and develop methods to improve the quality of a communication system [1]. These results can also be used to improve the performance of sonar systems [2], [3]. The shallow water acoustic (SWA) channel is one of the most challenging communication channels because it suffers the path-dependent Doppler and angle spreading, which results in a time-varying wideband channel impulse response with long delay spreads [4], [5]. While there exist several deterministic and statistical simulation models for SWA channels [6]-[11], a statistical framework for SWA channels, i.e., correlation functions, Doppler power spectral density, delay cross-power spectral density, etc., is still an open problem. These statistics are necessary tools for a proper design and analysis of multiple-input multiple-output (MIMO) mobileto-mobile (M-to-M) SWA communication or sonar systems. Paper approved by Prof. Michel Yacoub, the Associate Editor for IEEE Transactions on Vehicular Technology. Manuscript received April 23, 2010; revised November 11, 2010; revised February 18, 2011. Copyright (c) 2011 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to [email protected]. Alenka G. Zaji´c is with the School of Computer Science, Georgia Inst. of Tech., Atlanta, GA 30332 USA

Furthermore, the closed-form expressions of these statistics can be useful for estimation of physical parameters such as angle spread, mean angles of departure and arrival, etc. Abdi and Guo [12] were the first to derive the frequency correlation function for time-invariant stationary SWA channels, by taking into account only macro-scattering effects. In contrast, this paper models a time-varying MIMO M-toM SWA channel by taking into account both, the macro- and micro-scattering effects. We first introduce a new geometrybased statistical model for wideband MIMO M-to-M SWA multipath fading channels. The proposed model characterizes the sound propagation in shallow water isovelocity environments by combining the deterministic ray-tracing theory with the statistical methods needed to characterize the random components of the propagation medium. From the statistical model, the corresponding MIMO time-frequency correlation function (tf-cf), MIMO Doppler power spectral density (Dpsd), and MIMO delay cross-power spectral density (dcpsd) for a shallow water isovelocity environment are derived. Finally, to illustrate the utility of the proposed model, we compare the correlation functions, D-psd, and dc-psd with those obtained from the measurements in [13], [14]. The results show close agreement between the derived and measured channel statistics. Furthermore, they show that the proposed model outperforms the models in [12] and [14]. The remainder of the paper is organized as follows. Section II introduces the geometry-based statistical model for wideband MIMO M-to-M SWA multipath fading channels. Section III derives the MIMO tf-cf, the D-psd, and the dcpsd for a shallow water isovelocity environment. Section IV compares the analytical and measured results for the tf-cf, the D-psd, and the dc-psd. Finally, Section V provides some concluding remarks. II. G EOMETRY- BASED S TATISTICAL M ODEL FOR MIMO M- TO -M SWA C HANNELS This paper considers a medium or long range MIMO communication system with Lt transmit and Lr receive transducers. The propagation occurs in shallow water environments with constant sound speed, i.e., sound energy propagates along the plane waves. The propagation is characterized with either line-of-sight (LoS) or non-line-of-sight (NLoS) conditions between the transmitter (Tx ) and the receiver (Rx ). The MIMO M-to-M SWA channel can be described by an Lr × Lt matrix H(t, τ ) = [hij (t, τ )]Lr ×Lt of the input-delay spread functions. Fig. 1 depicts LoS path and several macro-eigenrays travelling between the Tx and Rx in a SWA channel with Lt =

2

Lr = 1 transducer elements. The SWA channel is modeled as a two-dimensional (2-D) waveguide bounded from the top and bottom. The surface and bottom boundaries reflect an acoustic signal, which results in multiple macro-eigenrays travelling between the Tx and Rx , as shown in Fig. 1. At any time instance t, the Rx receives 2S downward arriving (DA) macroeigenrays (i.e., the last reflection is from the surface), each one having different number of s surface and ˆb bottom reflections, where S denotes the maximum number of interactions between any DA macro-eigenray and the surface, 1 ≤ s ≤ S, and s − 1 ≤ ˆb ≤ s. For example, if a DA macro-eigenray has only one interaction with the surface, i.e., S = 1, there are two possible paths that this eigenray travelled. The first path is a single-bounced path, where a DA macro-eigenray started upwards, interacted with the surface, and arrived at the Rx , i.e., s = 1, ˆb = 0. The second path is a double-bounced path, where a DA macro-eigenray started downwards, interacted with the bottom, then interacted with the surface, and arrived at the Rx , i.e., s = 1, ˆb = 1. Note that for S = 1, both macro-eigenrays will reach the Rx because they have similar energy [11], [15]. Similarly, there are 2B upward arriving (UA) macro-eigenrays (i.e., the last reflection is from the bottom) with b bottom and sˆ surface reflections, where B denotes the maximum number of interactions between a UA macro-eigenray and the bottom, 1 ≤ b ≤ B and b − 1 ≤ sˆ ≤ b. In contrast to fixed-to-mobile cellular radio channels, where single-bounced eigenrays are dominant [16], [17], in SWA channels, the energy stays trapped in the waveguide and the number of bounces (i.e., interactions with the surface and bottom) can be large [15]. However, the experimental results for medium and long range SWA channels show that the number of different macro-eigenrays that arrive at the Rx rarely exceeds 8 i.e., 2S + 2B = 8 [11], [13], [14].

of the deterministic macro-eigenrays, there are random signal fluctuations (micro-eigenrays), which account for the time variability of the channel response [9], [15]. To implement this propagation mechanism, each DA macro-eigenray is modeled as an average of Nsˆb DA micro-eigenrays. Similarly, each UA macro-eigenray is modeled as an average of Mbˆs UA microeigenrays. Note that the possible positions of micro-scatterers are clustered around the positions of macro-scatterers, as shown in Fig. 2. This is in contrast to radio channel propagation, where scatterers can be randomly placed anywhere in a 2-D or 3-D plane.

Fig. 2. The geometry-based model for wideband MIMO M-to-M SWA channels with Lt = Lr = 2 transducer elements. Each macro-eigenray is represented as a large number of micro-eigenrays.

Fig. 2 shows a SWA channel with Lt = Lr = 2 transducer elements. This elementary 2 × 2 transducer configuration will be used later to construct uniform linear arrays with an arbitrary number of transducer elements. The horizontal spacing between the Tx and Rx is denoted by R. The water depth is denoted by h, while the depths of Tx and Rx are denoted by hT and hR , respectively. As we are interested in medium and long range shallow water communications, we assume that the depths h, hT , and hR are much smaller than the distance R. The spacing between two adjacent transducer elements at the Tx and Rx is dT and dR , respectively. It is assumed that dT and dR are much smaller than the depths h, hT , and hR . (S) (p) (p) (p) The symbols ²T sˆbn and ²T bˆsm denote distances Tx -Ssˆbn and (p)

Fig. 1. Illustration of LoS path and several macro-eigenrays travelling between the Tx and Rx in a SWA channel with Lt = Lr = 1 transducer elements.

Here we note that the exact positions of scatterers depend on the surface and bottom characteristics and may vary from one location to another and from one time instance to another. The roughness of sea surface and sea bottom is characterized by micro-scatterers, as shown in Fig. 2. On the other hand, the locations of macro-scatterers depend only on the waveguide geometry and the number of macro-eigenrays, and can be computed [15]. In other words, associated with each

(B)

(B)

(S)

Tx -Sbˆsm , respectively, where Ssˆbn and Sbˆsm denote the nth and mth micro-scatterers located around the macro-scatterers (S) (B) Ssˆb and Sbˆs at the surface and bottom, respectively, for 1 ≤ n ≤ Nsˆb and 1 ≤ m ≤ Mbˆs , as shown in Fig. 2. Similarly, (p) ˜ (p) ˜ (q) (q) (˜ q) (˜ q) the symbols ²T sˆbn , ²T bˆsm , ²Rsˆbn , ²Rbˆsm , ²Rsˆbn , ²Rbˆsm , ²pq , (p) ˜

(S)

(p) ˜

(B)

(S)

²pq , Tx -Sbˆsm , Ssˆbn ˜ , ²p˜ q , and ²p˜ ˜q denote distances Tx -Ssˆ bn (q)

(B)

(p)

(˜ q)

(q)

(S)

(˜ q)

(B)

(˜ q)

(p)

(q)

(p) ˜

(q)

Rx , Sbˆsm -Rx , Ssˆbn -Rx , Sbˆsm -Rx , Tx -Rx , Tx -Rx , (p) ˜

(˜ q)

Tx -Rx , and Tx -Rx , respectively. For ease of reference, Fig. 3 details the geometry of the LoS path as well as the geometry of single-bounced surface and single-bounced bottom micro-eigenrays (i.e., s = 1, ˆb = 0, b = 1, sˆ = 0) (S) (B) scattered from the Ss=1,ˆb=0,n -th and Sb=1,ˆs=0,m -th microscatterers, respectively. The geometry of multiple-bounced micro-eigenrays is similarly defined, but omitted from Fig. 3 for ease of reference. Angles θT and θR in Fig. 3 describe

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the orientations of Tx and Rx transducer arrays in the x-z plane, respectively, relative to the x-axis. The Tx and Rx are moving with constant speeds vT and vR in directions described by angles γT and γR in the x-z plane (relative to the x-axis), (p) (p) respectively. The symbols αT sˆbn and αT bˆsm are the angles of (p)

departure (AoD) of micro-eigenrays that start from Tx and (S) (B) impinge on the scatterers Ssˆbn and Sbˆsm , respectively, whereas (q)

(q)

αRsˆbn and αRbˆsm are the angles of arrival (AoA) of the micro(S)

(B)

(q)

eigenrays scattered from Ssˆbn and Sbˆsm and arriving at Rx , respectively. Finally, the symbols αT R and ²T R denote the AoA of LoS ray and the distance OT -OR , respectively. For ease of reference, the parameters defined in Figs. 1 - 3 are summarized in Table I.

spherical spreading and absorption, respectively, and δ(·) de(p,q) notes the Dirac impulse function. The LoS time delay τT R (t) denotes the travel time of the LoS path, i.e., vT ²pq (p,q) + t cos(αT R + π − γT ) τT R (t) = c c vR + t cos(αT R − γR ), (3) c where c is the speed of sound. We assume that the Tx has omnidirectional transducer elements and therefore produces a spherical wavefront in an isovelocity medium. The propagation loss caused by the spherical spreading can be written as LS (D) = 1/D [15]. Furthermore, when sound propagates in the ocean, part of the acoustic energy is continuously transformed into heat. This absorption is primarily a result of relaxation processes in seawater. The absorption loss can be written as [15] LA (D) = 10−Dβ/20000 , where

µ

β

Fig. 3. The detailed geometry of the LoS path, single-bounced surface and (B) (S) single-bounced bottom micro-eigenrays scattered from the S ˆ and Sb,ˆs,m s,b,n micro-scatterers, respectively.

Observe from the geometrical model in Fig. 2 that propagation through SWA channel can be characterized as a superposition of surface-bottom bounced macro-eigenrays and the LoS path between the Tx and Rx . The macro-eigenrays can be grouped into the UA macro-eigenrays (i.e., the last reflection is from the bottom) and the DA macro-eigenrays (i.e., the last reflection is from the surface). Furthermore, each DA macro-eigenray is represented as an average of Nsˆb DA micro-eigenrays, while each UA macro-eigenray is represented as an average of Mbˆs UA micro-eigenrays. Then, the input (q) (p) delay-spread function of the pressure link Tx -Rx can be written as a superposition of the LoS, UA, and DA macroeigenrays, viz. hpq (t, τ ) =

UA DA hLoS pq (t, τ ) + hpq (t, τ ) + hpq (t, τ ). (1)

The LoS component of the input delay-spread function is r q K LoS hpq (t, τ ) = Gp (αT R + π) Gq (αT R ) K+ ³1 ´ (p,q) LS (²T R )LA (²T R )δ τ − τT R (t) , (2) where Gp (·) and Gq (·) denote the radiation patterns of the pth transmit and q th receive transducer element, respectively, K is the Rice factor (ratio of LoS to scatter received power), LS (²T R ) and LA (²T R ) denote the propagation loss due to

(4)

=

Sl · A · fT · fc2 B · f2 + fT2 + fc2 fT −4 (1 − 6.54 · 10 · P ) [dB/km],

¶

8.68 · 103

(5)

and A = 2.34 · 10−6 , B = 3.38 · 10−6 , Sl is salinity (in %◦), P is hydrostatic pressure [kg/cm2 ], fc is the carrier frequency [kHz], fT = 21.9 · 106−1520/(T +273) is relaxation frequency [kHz], and T is the temperature [◦ C]. The upward and downward arriving components of the input delay-spread function are, respectively, r ηB A hU (t, τ ) = (6) pq 2B(K + 1) r B b X X 1 UA UA UA b LS (Dbˆ )L (D )L (θ ) A B bˆ s bˆ s s Mbˆs b=1 sˆ=b−1 r ³ M bˆ s ´ ³ ´ ³ ´ X (p) (q) (p,q) Gp αT bˆsm Gq αRbˆsm ξbˆsm ejφbˆsm δ τ − τbˆsm (t) , m=1

r

hDA pq (t, τ ) S s X X s=1 ˆ b=s−1 Nsˆb X n=1

=

ηS 2S(K + 1)

(7) s ˆ

)LA (DsDA )LB (θsDA )b LS (DsDA ˆ ˆ ˆ b b b

1 Nsˆb

r

³ ´ ³ ´ ³ ´ (p) (q) (p,q) Gp αT sˆbn Gq αRsˆbn ξsˆbn ejφsˆbn δ τ − τsˆbn (t) ,

UA DA where LS (Dbˆ ) denote the macro-eigenray s ) and LS (Dsˆ b UA propagation losses caused by spherical spreading, LA (Dbˆ s ) DA and LA (Dsˆb ) denote the macro-eigenray propagation losses UA DA caused by absorption, and Dbˆ denote the total s and Dsˆ b distances travelled by UA and DA macro-eigenrays with sˆ, b and s, ˆb surface-bottom interactions, respectively. The DA UA are obtained using the method of distances Dbˆ s and Dsˆ b images and can be written as [15] q UA Dbˆ = R2 + (2ˆ sh − hR + (−1)(b−ˆs) hT )2 , (8) s q DsDA = R2 + (2ˆbh + hR − (−1)(s−ˆb) hT )2 . (9) ˆ b

4

TABLE I D EFINITION OF THE PARAMETERS USED IN THE GEOMETRY- BASED MODEL FOR MIMO M- TO -M SWA CHANNELS . The horizontal spacing between the Tx and Rx. The water depth, the Tx depth, and the Rx depth, respectively. The spacing between two adjacent transducer elements at the Tx and Rx, respectively. The orientation of the Tx and Rx transducer array in the x-z plane (relative to the x-axis), respectively. The velocities of the Tx and Rx, respectively. The moving directions of the Tx and Rx, in the x-z plane (relative to the x-axis), respectively. The angles of departure (AoD) of the waves that impinge on

R h , hT , hR dT , d R

θT , θ R vT , v R γT , γ R ( p) α ( pˆ) , α Tb sˆm Tsbn

(q)

the scatterers S ( Sˆ ) and S b( sBˆm) , respectively. sbn

( q) α ( q)ˆ , α Rb sˆm

The angles of arrival (AoA) of the waves scattered from S ( Sˆ ) sbn

α TR , ε TR

and S b( sBˆm) , respectively. The AoA of line-of-sight path and the distance OT − O R , respectively.

Rsbn

~

( p) ε ( pˆ) , ε Tb , ε ( pˆ) , sˆm

Tsbn Tsbn ( ~p ) (q ) ε Tb , ε (q )ˆ , ε Rb , sˆm sˆm Rsbn (q~ ) ( q~ ) ε ˆ , ε Rbsˆm Rsbn ε pq , ε p~q , ε pq~ , ε p~q~ , UA ε DAˆ , ε Tbˆ s Tsb

,

ε DAˆ , Rsb

ε UA Rbˆs

( ) ( d (T , S ), d (S , R ), d (S and, d (S , R ) , respectively.

) ( ), d (S

~

) )

) The distances d T x( p ) , S ( Sˆ ) , d T x( p ) , S b( B , d T x( p ) , S ( Sˆ ) , sˆm p) (~ x

(B) bsˆm ( B) bsˆm

(S ) sbˆn

( q~ ) x

(

sbn ( q) x

( B) , R (q ) bsˆm x

s bn ~ (S ) (q ,R ) sbˆn x

~ ) ( ~ ) ( ) (B) (S ) ) d (OT , Sbsˆ ), d (S sbˆ , OR )

The distances d T x( p ) , R x(q ) , d T x( p ) , R x( q) , d T x( p ) , Rx( q ) ,

d

(

)

~ ~ T x( p ) , R x( q ) ,

(

)

(

d OT , S ( Sˆ ) , sb

and, d S b( sBˆ ) , O R , respectively.

The parameters ηS and ηB in (6) and (7), respectively, specify how much the UA and DA macro-eigenrays contribute in the total power, i.e., these parameters satisfy ηS + ηB = 1. The UA ) and LB (θbˆ parameters LB (θsDA s ) denote the impedance ˆ b mismatch between the seawater and seabed and can be written as [15] ¯ ¯ ¯ x cos θ − py 2 − sin2 θ ¯ ¯ ¯ p LB (θ) = ¯ (10) ¯, ¯ x cos θ + y 2 − sin2 θ ¯ where x = ρ1 /ρ, y = c/c1 , ρ and c denote the density and sound speed in the seawater, respectively, whereas ρ1 and c1 denote the density and sound speed in the seabed, respectively. UA DA are obtained as follows: The incidence angles θbˆ s and θsˆ b µ ¶ R UA −1 θbˆs = tan , (11) 2ˆ sh − hR + (−1)(b−ˆs) hT Ã ! R θsDA = tan−1 . (12) ˆ b ˆ 2bh + hR − (−1)(s−ˆb) hT Finally, ξbˆsm > 0 and ξsˆbn > 0 denote the amplitudes of micro-eigenrays, φbˆsm and φsˆbn denote the phases of micro(p,q) (p,q) eigenrays, and the time delays τbˆsm (t) and τsˆbn (t) are the travel times of UA and DA micro-eigenrays, respectively, i.e., (p,q)

τbˆsm (t) =

UA Dbˆ s c

(p)

(13) (q)

A ²U ²U A − ²Rbˆsm 1 (q) T bˆ s − ²T bˆ sm − Rbˆs + ∆Zbˆsm (t) sin αRbˆsm c c c i 1h (p) (q) + vT t cos(αT bˆsm − γT ) + vR t cos(αRbˆsm − γR ) , c DDA ˆ (p,q) τsˆbn (t) = sb (14) c (q) (p) ²DAˆ − ²Rsˆbn ²DAˆ − ²T sˆbn 1 (q) − Rsb + ∆Zsˆbn (t) sin αRsˆbn − T sb c c c

−

i 1h (p) (q) vT t cos(αT sˆbn − γT ) + vR t cos(αRsˆbn − γR ) , c where ∆Zsˆbn (t) and ∆Zbˆsm (t) denote the vertical displacements of surface micro-scatterers due to surface motion and A UA the distances ²DA , ²DAb , ²U T bˆ s , and ²Rbˆ s are defined in Table I. T sˆ b Rsˆ The travel times of UA and DA micro-eigenrays take into account movement of the Tx and Rx , the vertical displacements of surface micro-scatterers due to surface motion, and the location of MIMO hydrophones. (p) (p) It is assumed that the AoDs (αT sˆbn and αT bˆsm ), the +

(q)

AoAs (αRsˆbn and αRbˆsm ), and the vertical displacements ∆Zsˆbn (t) and ∆Zbˆsm (t) are random variables. Furthermore, it is assumed that the phases φsˆbn and φbˆsm are uniform random variables on the interval [−π, π) that are independent from the AoDs, the AoAs, and the vertical displacements of the surface. Here, we note that the locations of micro-scatterers that contribute to one macro-eigenray are independent from the locations of micro-scatterers that contribute to a different macro-eigenray because different macro-eigenrays have different angles of departure and arrival. Using the assumptions introduced above and the Central Limit Theorem [17], we can conclude that each macro-eigenray is an independent complex Gaussian process with zero mean. Sum of several complex Gaussian processes with zero means results in a new A complex Gaussian process with zero mean. Hence, hU pq (t, τ ) DA and hpq (t, τ ) are also independent complex Gaussian random (p) processes with zero means. Note that the AoDs (αT sˆbn and (p)

(q)

(q)

αT bˆsm ) are dependent on the AoAs (αRsˆbn and αRbˆsm ), respectively. Assuming small angle spreads, dT ¿ min(hT , h − hT ), and dR ¿ min(hR , h − hR ), we can approximate the (p) (p) ˜ AoDs and AoAs in Fig. 3 as αT sˆbn ≈ αT sˆbn ≈ αT sˆbn , (p)

(p) ˜

(q)

(˜ q)

αT bˆsm ≈ αT bˆsm ≈ αT bˆsm , αRsˆbn ≈ αRsˆbn ≈ αRsˆbn , and (q)

(˜ q)

αRbˆsm ≈ αRbˆsm ≈ αRbˆsm , where αT sˆbn , αT bˆsm , αRsˆbn , and αRbˆsm are defined in Fig. 3. Assuming that each microeigenray, when interacting with the surface and bottom, has equal incident and reflecting angles, we can observe that αT sˆbn = π − αRsˆbn and αT bˆsm = 3π − αRbˆsm . (p) (p) (q) (q) The distances ²T sˆbn , ²T bˆsm , ²Rsˆbn , ²Rbˆsm , and ²pq can be expressed as functions of the random AOAs αRsˆbn and αRbˆsm and the LoS angle αT R as follows hT Lt + 1 − 2p + dT cos(αRsˆbn + θT ) (15) sin αRsˆbn 2 h − hT Lt + 1 − 2p (p) ²T bˆsm ≈ − + dT cos(αRbˆsm + θT )(16) sin αRbˆsm 2 hR Lr + 1 − 2q (q) ²Rsˆbn ≈ − dR cos(αRsˆbn − θR ) (17) sin αRsˆbn 2 Lr + 1 − 2q h − hR (q) − dR cos(αRbˆsm −θR )(18) ²Rbˆsm ≈ − sin αRbˆsm 2 Lt + 1 − 2p ²pq ≈ ²T R + dT cos(αT R − θT ) − 2 Lr + 1 − 2q dR cos(αT R − θR ) (19) 2 p where ²T R = R2 + (hT − hR )2 and parameters p and q take values from the sets p ∈ {1, . . . , Lt } and q ∈ {1, . . . , Lr }, respectively. The derivations of approximations in (15)-(19) (p)

²T sˆbn ≈

5

are presented in Appendix A. To simplify further analysis, we normalize all time delays with respect to the LoS arrival time. This normalization is reflected in the distances ²pq , DsDA , and ˆ b UA Dbˆ , and they can be rewritten as s ²˜pq ˜ DA D sˆ b UA ˜ bˆ D s

= 0.5(Lt + 1 − 2p)dT cos(αT R − θT ) − 0.5(Lr + 1 − 2q)dR cos(αT R − θR ), (20) q = R2 + (2ˆbh + hR − (−1)(s−ˆb) hT )2 p R2 + (hT − hR )2 , (21) − q 2 (b−ˆ s ) 2 = R + (2ˆ sh − hR + (−1) hT ) p R2 + (hT − hR )2 . (22) −

time-variant transfer function is the Fourier transform of the input delay-spread function [17] and can be written as Tpq (t, f ) = =

Fτ {hpq (t, τ )} LoS UA DA Tpq (t, f ) + Tpq (t, f ) + Tpq (t, f )(29)

LoS UA DA where Tpq (t, f ), Tpq (t, f ), and Tpq (t, f ) are LoS, UA, and DA components of the time-variant transfer function, respectively. Using (2)-(24), the LoS, UA, and DA components of the time-variant transfer function in (29) can be written as r © LoS ª K LoS LS (²T R )LA (²T R ) Tpq (t, f ) = Fτ hpq (t, τ ) = K +1 2π ej c (fc +f )[vT t cos(αT R −γT )−vR t cos(αT R −γR )] (30) 2πfc

Since the depths h, hT , and hR are much smaller than the ej c [(0.5Lt +0.5−p)dT cos(αT R −θT )−(0.5Lr +0.5−q)dR cos(αT R −θR )] r B b ˜ DA and D ˜ U A in (21) and (22) can X X distance R, the distances D © UA ª ηB bˆ s UA √ sˆb be approximated using 1 + x ≈ 1 + x/2, for small x, as Tpq (t, f ) = Fτ hpq (t, τ ) = 2B(K + 1) b=1 sˆ=b−1 follows r M bˆ s X 1 UA UA UA b ˜ DA D (23) LS (Dbˆ ξbˆsm ejφbˆsm (31) ˆ ≈ s )LA (Dbˆ s )LB (θbˆ s ) Mbˆs m=1 i hsb ˆ h i 2 ˆb2 h2 + ˆbhhR − (−1)(s−b)ˆbhhT + (s − ˆb)hT hR /R, hR +hT −2h hR +hT −2h 2π(fc +f ) ˜ U A Dbˆ + sin −j s − sin µU Abˆ c αRbˆ s s m e UA ˜ bˆ 2π(fc +f ) D ≈ (24) hs i ej c vT t cos(αRbˆsm +γT ) 2 2 (b−ˆ s) 2π(fc +f ) 2 sˆ h − sˆhhR + (−1) sˆhhT + (b − sˆ)hT hR /R. e−j c [vR t cos(αRbˆsm −γR )+∆Zbˆsm (t) sin αRbˆsm ] 2πfc Lt +1−2p Lr +1−2q A UA ej c [ 2 dT cos(αRbˆsm +θT )− 2 dR cos(αRbˆsm −θR )] , ²DAb , ²U Finally, we note that ²DA T bˆ s , and ²Rbˆ s can be written T sˆ b Rsˆ r S s X X as functions of the mean AoA angles µDAsˆb and µU Abˆs , i.e., ª © DA ηT DA (t, τ ) = (t, f ) = F h T UA DA DA τ pq pq ²T sˆb = hT / sin µDAsˆb , ²Rsˆb = hR / sin µDAsˆb , ²T bˆs = −(h − 2T (K + 1) s=1 ˆ A b=s−1 hT )/ sin µU Abˆs , and ²U = −(h − h )/ sin µ , where the R U Abˆ s Rbˆ s s Nsˆb mean AoA angles µDAsˆb and µU Abˆs are depicted in Fig. 1. X 1 b DA ˆ DA DA ξ ˆ ejφsˆbn (32) The angles of arrival αRsˆbn and αRbˆsm are modeled using LS (Dsˆb )LA (Dsˆb )LB (θsˆb ) Nsˆb n=1 sbn the following Gaussian probability density functions (pdfs) · ¸ 2π(fc +f ) ˜ DA − hR +hT + hR +hT 1 D −j c sin µ sin α sˆ b DAsˆ b Rsˆ bn ftop (αRsˆbn ) = q e 2 2πσDAs 2π(fc +f ) ˆ b ej c vT t cos(αRsˆbn +γT ) 2 2π(fc +f ) )}, 0 < α < π, (25) exp{−(αRsˆbn − µDAsˆb )2 /(2σDAs ˆ ˆ Rsbn b e−j c [vR t cos(αRsˆbn −γR )+∆Zsˆbn (t) sin αRsˆbn ] Lr +1−2q 2πfc Lt +1−2p 1 fbottom (αRbˆsm ) = p ej c [ 2 dT cos(αRsˆbn +θT )− 2 dR cos(αRsˆbn −θR )] , 2 2πσU Abˆs 2 where fc denotes the carrier frequency. exp{−(αRbˆsm − µU Abˆs )2 /(2σU sm < 2π,(26) Abˆ s )}, π < αRbˆ where µDAsˆb and µU Abˆs are the mean AoAs, whereas σDAsˆb and σU Abˆs are the angle spreads. The vertical displacements ∆Zsˆbn (t) and ∆Zbˆsm (t) are modeled as zero-mean Gaussian random processes with stationary and independent increments and can be written as 2 1 −(∆Zsˆbn (t))2 /(2tζ∆Z ) DAsˆ b (27) f (∆Zsˆbn (t)) = q e 2 2πtζ∆Z ˆ DAsb

1

f (∆Zbˆsm (t)) = q

2 2πtζ∆Z U Abˆ s

2

e−(∆Zbˆsm (t))

2 /(2tζ∆Z

U Abˆ s

)

III. T IME -F REQUENCY C ORRELATION F UNCTION , D OPPLER P OWER S PECTRAL D ENSITY, AND D ELAY C ROSS -P OWER S PECTRAL D ENSITY OF THE MIMO M- TO -M SWA S TATISTICAL M ODEL Assuming an isovelocity shallow water environment, we first derive the tf-cf of the MIMO geometry-based statistical model. From the tf-cf, we derive the corresponding D-psd and dc-psd and discuss some simulation results for the tf-cf, D-psd and dc-psd.

(28)

2 2 where tζ∆Z and tζ∆Z denote the variances. U Abˆ s DAsˆ b To simplify further analysis, we use the time-variant transfer function instead of the input delay-spread function and we normalize the gain patterns of the transducer elements to unity (i.e., we assume omnidirectional array elements), although other gain patterns can be accommodated at this point. The

A. MIMO Time-Frequency Correlation Function The normalized tf-cf between two time-variant transfer functions Tpq (t1 , f1 ) and Tp˜ ˜q (t2 , f2 ), is defined as [17] Rpq,p˜ (33) ˜q (t1 , f1 , t2 , f2 ) = ∗ E [Tpq (t1 , f1 ) Tp˜ ˜q (t2 , f2 )] p , 2 Var[|Tpq (t1 , f1 )|2 ]Var[|Tp˜ ˜q (t2 , f2 )| ]

6

2πfc (p − p˜)dT sin(µU Abˆs + θT ) + c 2π(f2 − f1 ) [vR t2 sin(µU Abˆs −γR )−vT t2 sin(µU Abˆs +γT )]+ c 2π(fc + f1 ) [vR (t2 − t1 ) sin(µU Abˆs − γR ) − c LoS Rpq,p˜ vT (t2 − t1 ) sin(µU Abˆs + γT )] + ˜q (t1 , f1 , t2 , f2 ) = Rpq,p˜ ˜q (t1 , f1 , t2 , f2 ) + · ¸2 UA DA Rpq,p˜ (t , f , t , f ) + R (t , f , t , f ), (34) 2π(fc + f1 ) ˜q 1 1 2 2 pq,p˜ ˜q 1 1 2 2 2 (t2 − t1 )ζ∆Z sin µU Abˆs cos µU Abˆs + U Abˆ s c LoS UA DA where Rpq, (t , f , t , f ), R (t , f , t , f ), and R 1 1 2 2 1 1 2 2 p˜ ˜q pq,p˜ ˜q pq,p˜ ˜q · ¸2 ¸2 ¾ 2π(f2 − f1 ) (t1 , f1 , t2 , f2 ) denote the normalized tf-cfs of the LoS, UA, 2 t2 ζ∆Z sin µ cos µ , (41) U Abˆ s U Abˆ s U Abˆ s and DA components, respectively, and are defined as c ¤ £ LoS s S LoS (t1 , f1 )∗ Tp˜ E Tpq ηS X X −j 2π(f2c−f1 ) D˜ DA ˜q (t2 , f2 ) DA LoS sˆ b e R (t , f , t , f ) ≈ (t , f , t , f ) = Rpq,p˜ , (35) 1 1 2 2 1 1 2 2 pq,p˜ ˜q ˜q 2S s=1 ΩLoS /(K + 1) ˆ £ UA ¤ b=s−1 UA E Tpq (t1 , f1 )∗ Tp˜ 2πfc ˜q (t2 , f2 ) UA ˜ T cos(µDAsˆb +θT )−(q−˜ q )dR cos(µDAsˆb −θR )] Rpq, (t , f , t , f ) = , (36) ej c [(p−p)d p˜ ˜q 1 1 2 2 ΩU A /(K + 1) 2π(fc +f1 ) [vT (t2 −t1 ) cos(µDAsˆb +γT )−vR (t2 −t1 ) cos(µDAsˆb −γR )] ¤ £ DA c ej DA (t1 , f1 )∗ Tp˜ E Tpq 2π(f2 −f1 ) ˜q (t2 , f2 ) DA [vT t2 cos(µDAsˆb +γT )−vR t2 cos(µDAsˆb −γR )] c Rpq,p˜ , (37) ej ˜q (t1 , f1 , t2 , f2 ) = ΩDA /(K + 1) h i (t2 −t1 )ζ 2

where (·)∗ denotes complex conjugate operation, E[·] is the statistical expectation operator, Var[·] is the statistical variance operator, p, p˜ ∈ {1, . . . , Lt }, and q, q˜ ∈ {1, . . . , Lr }. Since UA DA Tpq (t, f ) and Tpq (t, f ) are independent zero-mean random processes, (33) can be simplified to

∆Z

−

DAsˆ b

2π(fc +f1 )

sin µ

2

DAsˆ b 2 c UA e where ΩLoS = (LS (²T R )LA (²T R ))2 , ΩU A = E[(LS (Dbˆ s ) 2 ½ · h i t ζ 2 UA UA b 2 DA σDAs LA (Dbˆ )LA (DsDA ) − 2 ∆ZDAsˆb 2π(f2 −f1 ) sin µ ˆ 2 ˆ s )LB (θbˆ s ) ) ], and ΩDA = E[(LS (Dsˆ ˆ b 2π(f2 − f1 ) b b DAsb 2 c e exp − ˆ LB (θsDA )b )2 ]. Under the wide sense stationary uncorrelated 2 c ˆ b scattering (WSSUS) condition, (34) simplifies to [17], [18] hR + hT 2πfc + (q − q˜)dR sin(µDAsˆb − θR ) − LoS c Rpq,p˜ (38) sin µDAsˆb tan µDAsˆb ˜q (∆t, ∆f ) = Rpq,p˜ ˜q (∆t, ∆f ) 2π(fc + f1 ) 2πfc UA DA + Rpq, (p − p˜)dT sin(µDAsˆb + θT ) + p˜ ˜q (∆t, ∆f ) + Rpq,p˜ ˜q (∆t, ∆f ), c ·c ¸ where ∆t = t2 − t1 and ∆f = f2 − f1 . By substituting (30) into (35), the expression for the MIMO vR (t2 −t1 ) sin(µDAsˆb −γR )−vT (t2 −t1 ) sin(µDAsˆb +γT ) + · ¸ tf-cf of LoS component becomes 2π(f2 − f1 ) vR t2 sin(µDAsˆb −γR )−vT t2 sin(µDAsˆb +γT ) + LoS c Rpq, (39) p˜ ˜q (t1 , f1 , t2 , f2 ) = K ¸2 · 2πfc 2π(fc + f1 ) 2 ej c [vT (t2 −t1 ) cos(αT R −γT )−vR (t2 −t1 ) cos(αT R −γR )] sin µDAsˆb cos µDAsˆb + (t2 − t1 )ζ∆ZDAsˆb 2πf1 c ej c [vT (t2 −t1 ) cos(αT R −γT )−vR (t2 −t1 ) cos(αT R −γR )] · ¸2 ¸2 ¾ 2π(f2 −f1 ) 2π(f2 − f1 ) [vT t2 cos(αT R −γT )−vR t2 cos(αT R −γR )] 2 c ej t2 ζ∆Z sin µ cos µ . (42) DAsˆ b DAsˆ b DAsˆ b 2πfc c ˜ T cos(αT R −θT )−(q−˜ q )dR cos(αT R −θR )] ej c [(p−p)d . Under WSSUS conditions, (41) and (42) simplify to For WSSUS conditions, (39) simplifies to

LoS Rpq, p˜ ˜q (∆t, ∆f ) = K

UA Rpq, p˜ ˜q (∆t, ∆f )

2πfc

ej c [vT ∆t cos(αT R −γT )−vR ∆t cos(αT R −γR )] 2πfc ˜ T cos(αT R −θT )−(q−˜ q )dR cos(αT R −θR )] ej c [(p−p)d .

b=1 sˆ=b−1

(40)

As shown in Appendix B, the MIMO tf-cfs of the UA and DA components can be closely approximated as UA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 ) ≈

B b UA ηB X X −j 2π(f2 −f1 ) D˜ bˆ s c e 2B b=1 sˆ=b−1

c j 2πf ˜ T cos(µU Abˆ q )dR cos(µU Abˆ s +θT )−(q−˜ s −θR )] c [(p−p)d

e 2π(fc +f1 ) [vT (t2 −t1 ) cos(µU Abˆ s +γT )−vR (t2 −t1 ) cos(µU Abˆ s −γR )] c ej ej e e

2π(f2 −f1 ) [vT t2 c

cos(µU Abˆ s +γT )−vR t2 cos(µU Abˆ s −γR )]

2 (t2 −t1 )ζ∆Z U Abˆ s − 2

−

2 t2 ζ∆Z U Abˆ s 2

h

h

2π(fc +f1 ) c

2π(f2 −f1 ) c

sin µU Abˆ s

sin µU Abˆ s

2 B b 2 ηB X X − ∆tζ∆Z2U Abˆs [ 2π s] λ sin µU Abˆ = e 2B

i2

½ · σ2 2π(f2 − f1 ) exp − U Abˆs 2 c

i2

2πfc hR + hT − 2h + (q − q˜)dR sin(µU Abˆs − θR )− sin µU Abˆs tan µU Abˆs c

2π

2π∆f

˜ UA

2π

Dbˆ ˜ T cos(µU Abˆ s +θT )−j s −γR ) s −j λ vR ∆t cos(µU Abˆ c ej λ (p−p)d 2π [−(q−˜ q )dR cos(µU Abˆ s −θR )+vT ∆t cos(µU Abˆ s +γT )] ej λ ½ · 2 σ 2π∆f hR + hT − 2h exp − U Abˆs + 2 c sin µU Abˆs tan µU Abˆs 2π [(q − q˜)dR sin(µU Abˆs −θR )−(p− p˜)dT sin(µU Abˆs +θT )]+ λ 2π [vR ∆t sin(µU Abˆs − γR ) − vT ∆t sin(µU Abˆs + γT )] + λ · ¸2 ¸2 ¾ 2π 2 ∆tζ∆Z sin µ cos µ , (43) U Abˆ s U Abˆ s U Abˆ s λ S s ∆tζ 2 2 ηS X X − ∆Z2DAsˆb [ 2π DA b] λ sin µDAsˆ Rpq, (∆t, ∆f ) = e p˜ ˜q 2S s=1 ˆ b=s−1

˜ DA −vR ∆t cos(µ j 2π q )dR cos(µDAsˆb −θR )−j 2π∆f D DAsˆ b −γR ) λ (q−˜ c sˆ b

e 2π ˜ T cos(µDAsˆb +θT )+vT ∆t cos(µDAsˆb +γT )] ej λ [−(p−p)d

7

· 2 σDAs hR + hT 2π ˆ b 2π∆f exp − + 2 c sin µDAsˆb tan µDAsˆb λ £ ¤ (q − q˜)dR sin(µDAsˆb −θR )−(p− p˜)dT sin(µDAsˆb +θT ) + ¤ 2π £ vR ∆t sin(µDAsˆb − γR ) − vT ∆t sin(µDAsˆb + γT ) + λ ¸2 ¸2 ¾ · 2πfc 2 ∆tζ∆ZDAsˆb sin µDAsˆb cos µDAsˆb , (44) c where λ denotes the carrier wavelength. From (39), (41), and (42) we can observe that the tf-cf of SWA channel follows an exponential function. This result is in agreement with the experimentally obtained tf-cfs [6], [13], [14]. Furthermore, this result differs from the traditional Bessel-shaped tf-cfs that are typical for radio channels. These differences are due to different propagation mechanisms in SWA and radio channels. Fig. 4 plots several single-input single-output (SISO) and MIMO tf-cfs as function of time, assuming WSSUS isovelocity scattering environment. We asNormalized TF Correlation |R(∆t,∆f=3 Hz,dT,dR)|

½

Rpq,p˜ ˜q (∆t, ∆f = 0) with respect to ∆t [17], [18]. From (34), it follows that the D-psd is a summation of the D-psd functions of the LoS, UA, and DA components, i.e., LoS UA DA Spq,p˜ ˜q (ν) = Spq,p˜ ˜q (ν) + Spq,p˜ ˜q (ν) + Spq,p˜ ˜q (ν).

By calculating the Fourier transform of the tf-cf in (40), we obtain the MIMO D-psd of the LoS component as follows LoS LoS Spq, p˜ ˜q (ν) = F∆t {Rpq,p˜ ˜q (∆t, ∆f = 0)} = 2π

˜ T cos(αT R −θT )−(q−˜ q )dR cos(αT R −θR )] j λ [(p−p)d Ke ³ ´ vR vT δ ν− cos(αT R − γR ) + cos(αT R − γT ) . (46) λ λ As shown in Appendix C, the D-psds of the UA and DA components are, respectively, UA Spq, p˜ ˜q (ν) =

TF-CF, dT=dR=0 TF-CF, dT=dR=0.5λ TF-CF, dT=dR=1λ

0.70 0.65 0.60

b=1 sˆ=b−1

2

2π ˜ T λ [(p−p)d

DA Spq, p˜ ˜q (ν) =

( e

√

π 2 0.5σU s Abˆ s AU Abˆ

A2 σ4 U Abˆ s U Abˆ s

ej

0.55 0.50

B b ηB X X − σU2 Abˆs BU2 Abˆs 2 e 2B

(j2πν+CU Abˆs +0.5σU2 Abˆs AU Abˆs BU Abˆs ) e

0.75

(45)

cos(µU Abˆ q )dR cos(µU Abˆ s +θT )−(q−˜ s −θR )] S X

ηS 2S s=1

s X

e

σ2 ˆ b − DAs 2

,

(47)

2 BDAs ˆ b

ˆ b=s−1

j2πν+C +0.5σ 2 A B DAsˆ b b DAsˆ b DAsˆ b DAsˆ σ4 A2 ˆ ˆ DAsb DAsb

2

)

√

π

2 A 0.5σDAs ˆ b b DAsˆ j 2π (p− p)d ˜ cos(µ +θ )−(q−˜ q )d cos(µ −θ T T R R )] DAsˆ b DAsˆ b λ [

0.45

e

0.40 0.35 0.30 0

10

20

30

40

50

60

70

80

90

100

Time ∆t [s]

Fig. 4. The magnitude of MIMO time-frequency correlation function as a function of time, for ∆f = 3 Hz and dT = dR = [0, 0.5λ, 1λ].

sume that the carrier frequency is fc = 1.7 kHz and the speed of sound is c = 1500 m/s. The distance between the Tx and Rx is set to R = 2 km, while the water, Tx , and Rx depths are h = 100 m, hT = 50 m, and hR = 50 m, respectively. The Tx and Rx have Lt = Lr = 2 vertically oriented elements (i.e., θT = θR = 90◦ ). It is assumed that the Tx and Rx are stationary, i.e., vT = vR = 0 m/s. The spacing between the Tx and Rx transducer elements are chosen from the set dT = dR ∈ [0, 0.5λ, 1λ]. For illustration purposes, the rest of the model parameters are chosen to be: K = 0.6, S = B = 1, ζ∆ZDA10 = ζ∆ZDA11 = ζ∆ZU A10 = ζ∆ZU A11 = 0.1, ηS = 0.6, µDA10 = 166◦ , µDA11 = 163.5◦ , µU A10 = 191.9◦ , µU A11 = 192.9◦ , σDA10 = 2.5◦ , σDA11 = 4◦ , σU A10 = 2.4◦ , σU A11 = 3◦ , αT R = 180◦ . The frequency deviation in Fig. 4 is set to ∆f = 3 Hz. The parameters used to obtain the curves in Fig. 4 are summarized in the forth column of Table II. We can observe that the tf-cf decreases as the spacings between the Tx and Rx elements, i.e., dT and dR , increase. B. MIMO Doppler Power Spectral Density The D-psd of the time-variant transfer function is the Fourier transform of the time-frequency correlation function

, (48)

where AU Abˆs , BU Abˆs , CU Abˆs , ADAsˆb , BDAsˆb , and CDAsˆb are · ¸2 2π 2 AU Abˆs = ζ∆ZU Abˆs sin µU Abˆs cos µU Abˆs + λ 2π [vR sin(µU Abˆs − γR ) − vT sin(µU Abˆs + γT )] (49) λ 2π BU Abˆs = − (p − p˜)dT sin(µU Abˆs + θT ) + λ 2π (q − q˜)dR sin(µU Abˆs − θR ), (50) λ · ¸2 2 ζ 2π CU Abˆs = ∆ZU Abˆs sin µU Abˆs + 2 λ 2π j [vR cos(µU Abˆs − γR ) − vT cos(µU Abˆs + γT )] , (51) λ · ¸2 2π 2 sin µDAsˆb cos µDAsˆb + ADAsˆb = ζ∆Z DAsˆ b λ ¤ 2π £ vR sin(µDAsˆb − γR ) − vT sin(µDAsˆb + γT ) (52) λ 2π BDAsˆb = − (p − p˜)dT sin(µDAsˆb + θT ) + λ 2π (q − q˜)dR sin(µDAsˆb − θR ), (53) λ · ¸ 2 2 ζ∆Z 2π DAsˆ b CDAsˆb = sin µDAsˆb + 2 λ ¤ 2π £ j vR cos(µDAsˆb − γR ) − vT cos(µDAsˆb + γT ) . (54) λ Fig. 5 shows several SISO and MIMO D-psds. We assume WSSUS isovelocity scattering environment with the carrier

8

C. MIMO Delay Cross-Power Spectral Density The dc-psd of the time-variant transfer function is the inverse Fourier transform of the time-frequency correlation function Rpq,p˜ ˜q (∆t = 0, ∆f ) with respect to ∆f [17], [18]. From (34), it follows that the dc-psd is a summation of the dc-psd functions of the LoS, UA, and DA components, i.e., LoS UA DA Ppq,p˜ ˜q (τ ) = Ppq,p˜ ˜q (τ ) + Ppq,p˜ ˜q (τ ) + Ppq,p˜ ˜q (τ ).

(55)

By calculating the inverse Fourier transform of the tf-cf in (40), we obtain the dc-psd of the LoS component as follows −1 LoS LoS Ppq, p˜ ˜q (τ ) = F∆f {Rpq,p˜ ˜q (∆t = 0, ∆f )} =

Kej

2π ˜ T λ [(p−p)d

cos(αT R −θT )−(q−˜ q )dR cos(αT R −θR )]

δ(τ ).(56)

It can be shown that the dc-psd of the UA and DA components are, respectively, UA Ppq, p˜ ˜q (τ ) =

B b ηB X X − σU2 Abˆs BU2 Abˆs 2 e 2B

(57)

b=1 sˆ=b−1

2π

˜ T cos(µU Abˆ q )dR cos(µU Abˆ s +θT )−(q−˜ s −θR )] ej λ [(p−p)d 2 ˜ U A /c−0.5σ 2 √ j2πτ −j2π D E B ( s U Abˆ s) bˆ s U Abˆ s U Abˆ π E2 σ4 U Abˆ s U Abˆ s e , 2 0.5σU s Abˆ s EU Abˆ S s 2 ˆ ηS X X − σDAs b B2 DA 2 DAsˆ b Ppq, (τ ) = (58) e p˜ ˜q 2S s=1 ˆ b=s−1

˜ T cos(µDAsˆb +θT )−(q−˜ q )dR cos(µDAsˆb −θR )] ej [(p−p)d 2 ˜ DA /c−0.5σ 2 j2πτ −j2π D E B √ ( b DAsˆ b) sˆ b DAsˆ b DAsˆ 4 2 π σ E ˆ ˆ DAsb DAsb , e 2 0.5σDAsˆb EDAsˆb 2π λ

where BU Abˆs and BDAsˆb are defined in (50) and (53), respectively, while EU Abˆs and EDAsˆb are defined as EU Abˆs =

2π hR + hT − 2h , c sin µU Abˆs tan µU Abˆs

(59)

EDAsˆb =

2π hR + hT . c sin µDAsˆb tan µDAsˆb

(60)

The derivations of (57) and (58) are omitted for brevity. Normalized Doppler Power Spectral Density [dB]

frequency fc = 10 kHz, and the speed of sound c = 1500 m/s. Furthermore, we choose the distance between the Tx and Rx to be R = 2 km. The water, Tx , and Rx depths are chosen to be h = 50 m, hT = 18 m, and hR = 18 m, respectively. The Tx and Rx have Lt = Lr = 2 transducer elements with orientations θT = θR = 90◦ . The Tx and Rx are moving with speeds vT = vR = 2 m/s and have orientations γT = γR = 40◦ . The spacing between the Tx and Rx transducer elements are chosen from the set dT = dR ∈ [0, 0.5λ, 1λ]. The rest of the model parameters are chosen to be: K = 0.6, S = B = 1, ζ∆ZDA10 = ζ∆ZDA11 = ζ∆ZU A10 = ζ∆ZU A11 = 0.09, µDA10 = 178◦ , µDA11 = 177◦ , µU A10 = 182◦ , µU A11 = 183◦ , σDA10 = 10◦ , σDA11 = 13.6◦ , σU A10 = 10◦ , σU A11 = 13.6◦ , αT R = 180◦ , ηS = 0.4, and ηB = 1−ηS . The parameters used to obtain the curves in Fig. 5 are summarized in the fifth column of Table II. From Fig. 5, we can observe that the D-psd magnitude and the Doppler spread decrease as the spacings between the Tx and Rx elements increase. On the other hand, the Doppler shift remains unchanged. Finally, we note that the D-psd of SWA channels differs from the U-shaped D-psd in cellular radio channels due to different propagation mechanisms.

0

D-psd, dT=dR=0 D-psd, dT=dR=0.5λ D-psd, dT=dR=1λ

-5

-10

-15

-20

-25

-30 -10

-8

-6

-4

-2

0

2

4

6

8

10

Doppler Frequency ν [Hz] Fig. 5. The normalized MIMO Doppler power spectral density for dT = dR = [0, 0.5λ, 1λ].

Fig. 6 plots several SISO and MIMO dc-psds, assuming WSSUS isovelocity scattering environment. We assume that the carrier frequency is fc = 17 kHz and the speed of sound is c = 1500 m/s. The distance between the Tx and Rx is set to R = 2 km, while the water, Tx , and Rx depths are h = 50 m, hT = 18 m, and hR = 18 m, respectively. The Tx and Rx have Lt = Lr = 2 vertically oriented elements (i.e., θT = θR = 90◦ ). It is assumed that the Tx and Rx are stationary, i.e., vT = vR = 0 m/s. The spacing between the Tx and Rx transducer elements are chosen from the set dT = dR ∈ [0, 1λ, 2λ]. The rest of the model parameters are set to: K = 0, S = B = 1, ζ∆ZDA10 = ζ∆ZDA11 = ζ∆ZU A10 = ζ∆ZU A11 = 0.1, µDA10 = 160◦ , µDA11 = 157◦ , µU A10 = 190◦ , µU A11 = 191◦ , σDA10 = 2.55◦ , σDA11 = 3.1◦ , σU A10 = 4◦ , σU A11 = 6◦ , αT R = 180◦ , ηS = 0.8, and ηB = 1 − ηS . The parameters used to obtain the curves in Fig. 6 are summarized in the sixth column of Table II. From Fig. 6, we can observe that the magnitude of dc-psd and the multipath spread decrease as the spacing between the Tx and Rx elements increase. In contrast to radio channels, where the later arriving multipaths carry less energy than the earlier arriving multipaths (i.e., the dc-psd decays exponentially), in SWA channels, it is often the case that the later arriving multipaths are received as stronger signals that the earlier arriving multipaths. This phenomenon can be explained by observing how macro-eigenrays propagate through the waveguide. Note that macro-eigenrays cannot be received with equal strength at all vertical locations in water because their angles of departure and arrival are predetermined by the waveguide geometry. For example, if the hydrophone is placed near the center of waveguide (water column), most of the arriving macro-eigenrays can be detected and the dcpsd decays exponentially (dT = dR = 0 in Fig. 6). As the hydrophones move away from the center of waveguide, some macro-eigenrays, although arriving, cannot be recorded at new vertical location, causing the imbalance in the dc-psd function (dT = dR = 2λ in Fig. 6).

Normalized Delay Cross-Power Spectral Density [dB]

9

TABLE II PARAMETERS USED IN F IGS . 4 - 11.

0 -1 -2

Definition

Type

Figs 4

Fig. 5

Fig.6

Fig.7Fig. 10

geometry-based

2

2

2

5

10

dT dR

The distance between the Tx and Rx. The spacing between two adjacent array elements at the Tx and Rx, respectively.

geometry-based

geometry-based

0, 1 λ , 2λ 0, 1 λ , 2λ 50 18 18

0 0.5 m

The water depth, the Tx depth, and the Rx depth, respectively.

0, 0.5 λ , 1λ 0, 0.5 λ , 1λ 50 18 18

0 0

h [m] hT [m] hR [m]

0, 0.5 λ , 1λ 0, 0.5 λ , 1λ 100 50 50

20 19 18

103 50 50

θT

The orientation of the Tx and Rx transducer array relative to the xaxis, respectively. The moving directions of the Tx and Rx, in the x-z plane (relative to the x-axis), respectively. The speed of sound The carrier frequency

geometry-based

90o

90o

90o

90o

90o

90o

90o

90o

90o

90o

90o

40o

90 o

90 o

90 o

90o

o

40

90 o

90 o

90 o

geometry-based geometry-based

1500 1.7

1500 10

1500 17

1470 17

1440 1.2

geometry-based or estimated

0 0

2 2

0 0

0.0006 0.00025

0 0

Estimated or selected to represent given propagation environment Estimated or selected to represent given propagation environment

0.6

0.6

0

0.8

0.71

0.1 0.1 0.1 0.1

0.09 0.09 0.09 0.09

0.1 0.1 0.1 0.1

0.0097 0.0097 0.0087 0.0087

0.019 0.019 0.015 0.015

Parameters

-3

R [km]

-4 -5 -6 -7 -8 -9 -10 -11

θR γT , γ R

dc-psd, dT=dR=0 dc-psd, dT=dR=1λ dc-psd, dT=dR=2λ

-12 -13 -14 -15 0

2

4

6

8

10

12

14

16

18

20

Delay τ [mS]

Fig. 6. The normalized MIMO delay cross-power spectral density for dT = dR = [0, 1λ, 2λ].

c [m/s] f c [kHz] vT [m/s] v R [m/s]

K

ζ ∆Z DA10

IV. C OMPARISON WITH M EASURED DATA In this section, we compare the theoretical results in Section III with the AUVFest07 measured data in [13], collected at calm sea. Furthermore, we compare the derived spatial correlation function Rpq,p˜ ˜q (∆t = 0, ∆f = 0) in (38) with the spatial correlation functions of the exponential model in [14], the particle-velocity model in [12], and the ASCOT01 measured data in [14]. The channel measurements in [13] are collected at fc = 17 kHz and the speed of sound is c = 1470 m/s. The distance between the Tx and Rx is R = 5 km. The water, Tx , and Rx depths are h = 20 m, hT = 19 m, and hR = 18 m, respectively. The Tx and Rx are equipped with vertically oriented transducer elements (i.e., θT = θR = 90◦ ). It is assumed that the Tx and Rx are relatively stationary, i.e., slightly moving with waves (i.e., γT = γR = 90◦ ). Figs. 7 and 8 compare the analytical temporal and frequency correlation functions, i.e., R11,11 (∆t, 0) and R11,11 (0, ∆f ) in (38), with the measured temporal and frequency correlation functions, respectively. Furthermore, Figs. 9 and 10 compare the analytical and measured D-psds and dc-psds, respectively. The analytical curves in Figs. 7 - 10 are obtained with the parameters K = 0.8, S = B = 1, vT = 0.0006 m/s, vR = 0.00025 m/s, µDA10 = 169◦ , µDA11 = 167◦ , µU A10 = 194◦ , µU A11 = 195◦ , σDA10 = 3.25◦ , σDA11 = 5◦ , σU A10 = 2.7◦ , σU A11 = 3.2◦ , ζ∆ZDA10 = ζ∆ZDA11 = 0.0097, ζ∆ZU A10 = ζ∆ZU A11 = 0.0087, ηS = 0.28, and ηB = 1 − ηS . The Rice factor K is estimated using the momentmethod in [20], and the parameters S and B are visually estimated from Fig. 3c in [13]. The rest of the parameters, i.e., [vT , vR , ζ∆ZDAsˆb , ζ∆ZU Abˆs , µDAsˆb , µU Abˆs , σDAsˆb , σU Abˆs , ηS ] are estimated jointly using the maximum-likelihood approach in [21] and the constraint ηS + ηB = 1. The maximumlikelihood function is optimized using the sequential quadratic programming algorithm [22], which is implemented in Matlab as a “fminsearch” function. The parameters used to obtain the curves in Figs. 7- 10 are summarized in the seventh column of Table II.

ζ ∆Z DA11 ζ ∆ZUA10

The velocities of the Tx and Rx, respectively. The Rice factor

The variances in the Gaussian pdf for the vertical displacements.

ζ ∆ZUA11 µ DA10 µ DA11 µ UA10 µ UA11

The mean AoAs.

σ DA10 σ DA11 σ UA10 σ UA11 ηS

The angle spreads for AoAs.

ηB

Specify how much the UA and DA rays contribute in the total averaged power, i.e., η S + η B =1

geometry-based

Estimated or selected to represent given propagation environment Estimated or selected to represent given propagation environment Estimated or selected to represent given propagation environment

Fig.11

166 o

178 o

160 o

169 o

173 .8o

163 .5o

177 o

157 o

167 o

173 .4o

o

o

o

o

190

194

183 o

191 .9

182

192 .9o

183 o

191 o

195 o

2.5o

10 o

2 .55o

3.25o

9.1o

4o 2.4o

13 .6o

3.1o

5o

10 .2o

10 o

4o

2.7 o

5.59 o

3o

13 .6o

6o

3.2 o

8.89o

0.6

0.4

0.8

0.28

0.44

0.4

0.6

0.2

0.72

0.56

184 .5o

The results in Figs. 7 - 10 show good match between the measured and analytical data. This result illustrates the validity of the derived statistics and validates the estimation methods used to find the model parameters. Furthermore, the results in Figs. 7 and 8 show that both temporal and frequency correlation functions decay exponentially with ∆t and ∆f , respectively. Finally, in contrast to radio channels where signals de-correlate relatively fast, in SWA channels signals stay highly correlated. The reason for this behavior is the reduced amount of randomness in SWA channel. Note that macro-eigenrays are deterministic and only randomness is introduced by micro-eigenrays that have small variations around macro-eigenrays. Finally, Fig. 11 compares the spatial correlation Rpq,p˜ ˜q (0, 0) in (38) with the analytical spatial correlation in [12] and the analytical and empirical spatial correlations in [14]. The ASCOT01 channel measurements are collected at the carrier frequency 1.2 kHz and the speed of sound is c = 1440 m/s (i.e., λ=1.2 m) [14]. The distance between the Tx and Rx is R = 10 km. The water, Tx , and Rx depths are h = 103 m, hT = 50 m, and hR = 50 m, respectively. The Rx is equipped with 33 vertically oriented transducer elements (i.e., θR = 90◦ ) with L = dR = 0.5 m element spacing, while the Tx is equipped with one vertical transducer (i.e., θT = 90◦ ). The vertical correlation is calculated with respect to eighth element from the bottom of the 33-element array. The analytical spatial

1.00

0.95

0.90

0.85

0.80

0.75

0.70 0

5

10

15

20

25

30

Time ∆t [s]

Frequency Correlation |R(∆t=0,∆f,dT=0,dR=0)|

Fig. 7. The magnitudes of analytical and measured temporal correlation functions. 1.00

Analytical Frequency Correlation Measured Frequency Correlation

0.95

0.90

0.85

0.80

Normalized Doppler Power Spectral Density [dB]

a large number of random macro-eigenrays. However, this is unrealistic assumption because the experimental results show that the number of different macro-eigenrays that arrive at the receiver rarely exceeds eight [11], [14], [13]. In contrast, the proposed model characterizes SWA channel as a superposition of a LoS component and several macro-eigenrays, where each macro-eigenray is modeled as a large number of microeigenrays. This approach leads to a closer match with the experimental data. Note that a large number of parameters helps fine-tuning the model rather than fitting the model to any measured curve, as can be observed from Table II.

Analytical Temporal Correlation Measured Temporal Correlation

0

Analytical D-psd Measured D-psd

-5 -10 -15 -20 -25 -30 -35 -40 -45 -1.2

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Doppler Frequency ν [Hz] 0.75

Fig. 9. density.

0.70 0

2

4

6

8

10

12

14

16

18

The normalized analytical and measured Doppler power spectral

20

Frequency ∆f [Hz]

Fig. 8. The magnitudes of analytical and measured frequency correlation functions.

correlation in (38) is obtained with the estimated parameters K = 0.71, S = B = 1, µDA10 = 173.8◦ , µDA11 = 173.4◦ , µU A10 = 183◦ , µU A11 = 184.5◦ , σDA10 = 9.1◦ , σDA11 = 10.2◦ , σU A10 = 5.59◦ , σU A11 = 8.89◦ , ηS = 0.44, ζ∆ZDA10 = ζ∆ZDA11 = 0.019, ζ∆ZU A10 = ζ∆ZU A11 = 0.015, and ηB = 1 − ηS . The parameters R, h, hT , hR , θT , θR , and αT R ≈ π, are selected to match the measurement conditions described above. The parameters used to obtain the curves in Fig. 11 are summarized in the eight column of Table II. For the reference, Fig. 11 also shows the spatial correlation functions of the exponential model in [14], i.e, exp{−L2 /(2λ)2 } and the particle-velocity model in [12] (with parameters Λb = 0.56, µb = 183◦ , µs = 173◦ , σb = 2.29◦ , and σs = 8◦ ). One can observe that the proposed model provides a closer match with the experimental correlation than the models in [12] and [14]. Although the proposed model has a large number of parameters that can be adjusted, they are not the reason for an improved fidelity to experimental data. The exponential model in [14] fits an exponential function with distance between hydrophones as a parameter into the measured data. On the other hand, the particle-velocity model in [12] models a channel impulse response in SWA channel as

The close agreement between the analytical and measured statistics in Figs. 7 - 11 confirms the utility of the derived statistics. Here we note that the variance of vertical displacement and the means and variances of AoAs can be characterized as functions of sea state (i.e., water depth, seafloor roughness, etc.). However, this is not a straight forward task and is outside of the scope of this paper. 0

Normalized Delay Cross-Power Spectrum [dB]

Normalized Temporal Correlation |R11,11(∆t,∆f=0)|

10

Analytical dc-psd Measured dc-psd

-2

-4

-6

-8

-10

-12 0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Delay τ [mS] Fig. 10. The normalized analytical and measured delay cross-power spectral density.

11

V. C ONCLUSIONS This paper proposed the geometry-based statistical model for MIMO M-to-M SWA fading channels. From the statistical model, the corresponding MIMO time-frequency correlation function, Doppler power spectral density, and delay crosspower spectral density are derived and illustrated with several examples. Finally, the derived characteristics are compared with the experimentally obtained channel characteristics and the close agreement is observed. The main advantage of the proposed model is that it expresses the acoustic field correlation as a function of the angles of arrival and angle spreads. This allows system engineers to understand how these channel parameters affect the correlation, which in turn provides useful guidelines for the system design.

¸2 µ ¶2 · h − hR (Lr + 1 − 2q) = dR + (A.4) 2 sin(αRbˆsm − π) µ ¶ µ ¶ Lr + 1 − 2q h − hR −2 dR cos (αRbˆsm −θR) , 2 sin(αRbˆsm −π) · ¸2 Lr + 1 − 2q ²2pq = dR +²2 − 2 Lr + 1 − 2q dR ² cos (αT R − θR) , (A.5) 2 2 (q) ²Rbˆsm

2

where ²2 = [(0.5Lt + 0.5 − p)dT ] + ²2T R − 2(0.5Lt + 0.5 − p)dT ²T R cos(θT − π − αT R ), and ²2T R = R2 + (hT − hR )2 . For small angle spreads, dT ¿ min(hT , h − h√ T ), and dR ¿ min(hR , h − hR ), we can use approximation 1 + x ≈ 1 + x/2, for small x to obtain the expressions in (15)-(19). A PPENDIX B D ERIVATION OF THE TF - CFS FOR UA AND DA C OMPONENTS

ACKNOWLEDGMENT

Spatial Correlation |R(∆t=0),∆f=0,dT=0,dR)|

The author would like to thank the anonymous reviewers whose feedback helped improve the quality of this paper. 1.0

Proposed Model Particle-velocity Model [12] Exponential Model [14] ASCOT01 Measured Data [14]

0.9 0.8 0.7

Substituting (31) and (32) into (36) and (37), respectively, and noting that the phases φbˆsm and φsˆbn are independent and uniformly distributed over [−π, π), the tf-cfs of UA and DA components can be written as UA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 ) =

0.6 0.5

−j

0.4 0.3 0.2 0.1 0.0 -8

-6

-4

-2

0

2

4

6

8

10

12

14

16

18

20

22

24

Array Element Number Fig. 11. Comparison of the analytical spatial correlations in (38), [12], and [14] with the measured spatial correlation in [14].

2π(f2 −f1 ) c

b=1 sˆ=b−1 h i ˜ U A − hR +hT −2h + hR +hT −2h D

bˆ s sin µU Abˆ sin αRbˆ s sm e j 2π [(p− p)d ˜ cos(α +θ )−(q−˜ q )d cos(α T T R Rbˆ sm Rbˆ sm −θR )] e λ 2π ej λ [vT (t2 −t1 ) cos(αRbˆsm +γT )−vR (t2 −t1 ) cos(αRbˆsm −γR )] 2πf1 ej c [vT (t2 −t1 ) cos(αRbˆsm +γT )−vR (t2 −t1 ) cos(αRbˆsm −γR )] 2π(f2 −f1 ) [vT t2 cos(αRbˆ sm +γT )−vR t2 cos(αRbˆ sm −γR )] c ej ·

E∆Zbˆsm (t) e−j e

−j

2π(fc +f1 ) (∆Zbˆ sm (t2 )−∆Zbˆ sm (t1 )) sin αRbˆ sm c

2π(f2 −f1 ) ∆Zbˆ sm (t2 ) sin αRbˆ sm c

DA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 )

A PPENDIX A D ERIVATIONS OF E QUATIONS (15) - (19) (p)

From Fig. 3, using the cosine law, the distances ²T sˆbn ,

(p) ²T bˆsm ,

(q) ²Rsˆbn ,

(q) ²Rbˆsm ,

and ²pq can be written as ¸2 µ ¶2 · hT (Lt + 1 − 2p) (p) 2 ²T sˆbn = dT + − (A.1) 2 sin(π − αRsˆbn ) µ ¶ µ ¶ ¡ ¢ Lt + 1 − 2p hT 2 dT cos θT −π +αRsˆbn , 2 sin(π −αRsˆbn ) · ¸2 µ ¶2 (Lt + 1 − 2p) h − hT (p) 2 ²T bˆsm = dT + −(A.2) 2 sin(αRbˆsm − π) ¶ µ ¶ µ h − hT Lt + 1 − 2p dT cos (θT −π +αRbˆsm ) , 2 2 sin(αRbˆsm − π) · ¸2 µ ¶2 (Lr + 1 − 2q) hR (q) 2 ²Rsˆbn = dR + − (A.3) 2 sin αRsˆbn µ ¶ µ ¶ ¡ ¢ Lr + 1 − 2q hR 2 dR cos αRsˆbn −θR , 2 sin αRsˆbn

Mbˆ B b s ηB X X 1 X 2 E[ξbˆ sm ] 2B Mbˆs m=1

−j

2π(f2 −f1 ) c

 ˜ DA − D sˆ b

¸ ,

(B.1)

Nsˆb S s ηS X X 1 X = E[ξs2ˆbn ] 2S s=1 Nsˆb n=1 ˆ b=s−1

hR +hT sin µ DAsˆ b



h +h + sinRα T Rsˆ bn

e 2π ˜ T cos(αRsˆbn +θT )−(q−˜ q )dR cos(αRsˆbn −θR )] ej λ [(p−p)d v (t −t ) cos(α +γ )−v j 2π [ T R (t2 −t1 ) cos(αRsˆ Rsˆ bn bn −γR )] e λ T 2 1 2πf1 ej c [vT (t2 −t1 ) cos(αRsˆbn +γT )−vR (t2 −t1 ) cos(αRsˆbn −γR )] 2π(f2 −f1 ) c

[vT t2 cos(αRsˆbn +γT )−vR t2 cos(αRsˆbn −γR )] · 2π(fc +f1 ) (∆Zsˆbn (t2 )−∆Zsˆbn (t1 )) sin αRsˆbn c E∆Zsˆbn (t) e−j ¸ 2π(f2 −f1 ) −j ∆Zsˆbn (t2 ) sin αRsˆbn c e , (B.2) ej

where E∆Zbˆsm (t) [·] and E∆Zsˆbn (t) [·] denote the statistical expectations with respect to ∆Zbˆsm (t) and ∆Zsˆbn (t), respectively. 2 2 ]/Nsˆb in (B.1) and (B.2) The terms E[ξbˆ s and E[ξsˆ sm ]/Mbˆ bn (B)

represent the average powers received from the scatterers Sbˆsm PMbˆs (S) 2 and Ssˆbn , respectively. Assuming that m=1 E[ξbˆ s = sm ]/Mbˆ

12

PNsˆb E[ξs2ˆbn ]/Nsˆb = 1, and the large number of micro1, n=1 eigenrays, i.e., Mbˆs À 1 and Nsˆb À 1, the discrete AoAs, αRbˆsm and αRsˆbn can be replaced with continuous random variables αRbˆs and αRsˆb characterized with pdfs fbottom (αRbˆs ) and ftop (αRsˆb ) respectively. Furthermore, the PMbˆs PNsˆb summations n=1 in (B.1) and (B.2) can be m=1 and replaced by integrals, i.e., UA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 ) = 2π(f2 −f1 ) c

Z B b X X

ηB 2B

−

2π

e

−j

e

j 2π ˜ T cos(αRbˆ q )dR cos(αRbˆ sm +θT )−(q−˜ sm −θR )] λ [(p−p)d

Rbˆ sm

ej

2π sm +γT )−vR (t2 −t1 ) cos(αRbˆ sm −γR )] λ [vT (t2 −t1 ) cos(αRbˆ

ej 2πf1 ej c [vT (t2 −t1 ) cos(αRbˆsm +γT )−vR (t2 −t1 ) cos(αRbˆsm −γR )] 2π(f2 −f1 ) [vT t2 cos(αRbˆ sm +γT )−vR t2 cos(αRbˆ sm −γR )] c ej · E∆Zbˆsm (t) e−j e−j

2π(fc +f1 ) (∆Zbˆ sm (t2 )−∆Zbˆ sm (t1 )) sin αRbˆ sm c

2π(f2 −f1 ) ∆Zbˆ sm (t2 ) sin αRbˆ sm c

DA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 )

ηS = 2S

· −j

2π(f2 −f1 ) c

¸ dαRbˆs ,

S X

s X

s=1 ˆ b=s−1

(B.3)

Z

2 (αRbˆ s −µU Abˆ s) 2

e

2π(f2 −f1 ) [vT t2 c

−j

Z 2π B b ηB X X 1 p (B.6) 2 2B 2πσU Abˆ s b=1 sˆ=b−1 π

2π(f2 −f1 ) c

h

˜ U A − hR +hT −2h + hR +hT −2h D bˆ s sin µ sin α

i

0

¸

Rsˆ bn

e 2π ˜ T cos(αRsˆbn +θT )−(q−˜ q )dR cos(αRsˆbn −θR )] ej λ [(p−p)d j 2π v (t −t ) cos(α +γ )−v [ T R (t2 −t1 ) cos(αRsˆ Rsˆ bn bn −γR )] e λ T 2 1 2πf1 ej c [vT (t2 −t1 ) cos(αRsˆbn +γT )−vR (t2 −t1 ) cos(αRsˆbn −γR )] 2π(f2 −f1 ) [vT t2 cos(αRsˆbn +γT )−vR t2 cos(αRsˆbn −γR )] c ej · 2π(fc +f1 ) (∆Zsˆbn (t2 )−∆Zsˆbn (t1 )) sin αRsˆbn c E∆Zsˆbn (t) e−j ¸ 2π(f2 −f1 ) −j ∆Zsˆbn (t2 ) sin αRsˆbn c e dαRsˆb , (B.4) Since ∆Zbˆs (t) has stationary and independent increments with pdf defined in (28), the expectation E∆Zbˆs (t) [·] in (B.3) can be calculated as follows Z ∞ 2π(fc +f1 ) (∆Zbˆ s (t2 )−∆Zbˆ s (t1 )) sin αRbˆ s c e−j (B.5) −∞

f (∆Zbˆs (t2 ) − ∆Zbˆs (t1 ))d(∆Zbˆs (t2 ) − ∆Zbˆs (t1 )) Z ∞ 2π e−j c sin αRbˆs (f2 −f1 )∆Zbˆs (t2 ) f (∆Zbˆs (t2 ))d(∆Zbˆs (t2 )) = −∞

· ¸2 2 (t2 − t1 )ζ∆Z 2π(fc + f1 ) U Abˆ s sin αRbˆs − exp − 2 c · ¸2 ¾ 2 t2 ζ∆Z 2π(f2 − f1 ) U Abˆ s sin αRbˆs . 2 c ½

cos(αRbˆ s +γT )−vR t2 cos(αRbˆ s −γR )]

2 (t2 −t1 )ζ∆Z U Abˆ s − 2

−

2 t2 ζ∆Z U Abˆ s 2

h

h

2π(fc +f1 ) c

2π(f2 −f1 ) c

sin αRbˆ s

sin αRbˆ s

i2

i2

dαRbˆs , Z π S s X X ηS 1 DA q (t , f , t , f ) = Rpq, (B.7) p˜ ˜q 1 1 2 2 2 2S s=1 0 2πσ ˆ b=s−1 DAsˆ b · ¸

e

−

π

˜ DA − hR +hT + hR +hT D sin µ sin α sˆ b DAsˆ b

UA Rpq, p˜ ˜q (t1 , f1 , t2 , f2 ) =

(2σ ) U Abˆ s Rbˆ s U Abˆ s e c [(p−p)d j 2πf ˜ cos(α +θ )−(q−˜ q )d cos(α −θ )] T T R R Rbˆ s Rbˆ s e c 2πfc ej c [vT (t2 −t1 ) cos(αRbˆs +γT )−vR (t2 −t1 ) cos(αRbˆs −γR )] 2πf1 ej c [vT (t2 −t1 ) cos(αRbˆs +γT )−vR (t2 −t1 ) cos(αRbˆs −γR )]

b=1 sˆ=b−1 π i h hR +hT −2h hR +hT −2h UA ˜ Dbˆ + sin s − sin µ α U Abˆ s

Substituting (B.5) and (26) into (B.3) and (B.6) and (25) into (B.4), we obtain the following expressions for the tf-cfs of the UA and DA components, respectively,

(α

−µ ) Rsˆ b DAsˆ b (2σ 2 ) DAsˆ b

2

−j

2π(f2 −f1 ) c

˜ DA − hR +hT + hR +hT D sin µ sin α sˆ b DAsˆ b

Rsˆ b

e e 2πfc ˜ T cos(αRsˆb +θT )−(q−˜ q )dR cos(αRsˆb −θR )] ej c [(p−p)d 2πfc ej c [vT (t2 −t1 ) cos(αRsˆb +γT )−vR (t2 −t1 ) cos(αRsˆb −γR )] 2πf1 ej c [vT (t2 −t1 ) cos(αRsˆb +γT )−vR (t2 −t1 ) cos(αRsˆb −γR )] 2π(f2 −f1 ) [vT t2 cos(αRsˆb +γT )−vR t2 cos(αRsˆb −γR )] c ej e

−

e

−

2 (t2 −t1 )ζ∆Z 2 2 t2 ζ∆Z DAsˆ b 2

DAsˆ b

h

h

2π(fc +f1 ) c

2π(f2 −f1 ) c

sin αRsˆb

sin αRsˆb

i2

i2

dαRsˆb .

For small angle spreads, the AoAs αRsˆb and αRbˆs are mainly concentrated around the mean AoAs µDAsˆb and µU Abˆs , respectively. Using the first-order Taylor expansion, the AoA angles can be approximated as follows cos(αRsˆb ) ≈ cos(µDAsˆb ) − sin(µDAsˆb )(αRsˆb − µDAsˆb )(B.8) sin(αRsˆb ) ≈ sin(µDAsˆb ) + cos(µDAsˆb )(αRsˆb − µDAsˆb )(B.9) sin(αRsˆb )2 ≈ sin(µDAsˆb )2 + 2 sin(µDAsˆb ) cos(µDAsˆb ) (αRsˆb − µDAsˆb ) (B.10) −1 −1 sin(αRsˆb ) ≈ sin(µDAsˆb ) − cos(µDAsˆb ) (αRsˆb − µDAsˆb )/ sin(µDAsˆb )−2 . (B.11) The similar approximations are applied to the AoAs αRbˆs . Using approximations and the equality R jαx these2 trigonometric 2 2 2 2 e (2πσ )−1/2 e−x /(2σ ) dx = e−σ α /2 [12], the tf-cfs of the UA and DA components can be written as in (41) and (42), respectively.

Similarly, the expectation E∆Zsˆb (t) [·] in (B.4) becomes ½ exp

−

2 t2 ζ∆Z

2

2 (t2 − t1 )ζ∆Z

DAsˆ b

·

2π(fc + f1 ) sin αRsˆb 2 c · ¸2 ¾ 2π(f2 − f1 ) sin αRsˆb . c DAsˆ b

¸2 −

A PPENDIX C T HE D- PSD OF THE UA AND DA C OMPONENTS The Fourier transforms of the time-frequency correlation UA DA functions Rpq, p˜ ˜q (∆t, ∆f = 0) and Rpq,p˜ ˜q (∆t, ∆f = 0) can

13

be written as UA Spq, p˜ ˜q (ν) =

ej

2π ˜ T λ [(p−p)d

ηB 2B

B X

b X

(C.1) b=1 sˆ=b−1

Z

cos(µU Abˆ q )dR cos(µU Abˆ s +θT )−(q−˜ s −θR )]

∞

−∞

½ · · ¸2 σ2 2π 2π 2 exp − U Abˆs ∆tζ∆Z sin µU Abˆs cos µU Abˆs + U Abˆ s 2 λ λ 2π (q − q˜)dR sin(µU Abˆs − θR ) + vR ∆t sin(µU Abˆs − γR ) − λ ¸2 ¾ [(p − p˜)dT sin(µU Abˆs + θT ) − vT ∆t sin(µU Abˆs + γT )] ∆tζ 2

2

∆ZU Abˆ s 2π sin µ [λ U Abˆ s] 2 e−j2πν∆t− 2π e−j λ [vR ∆t cos(µU Abˆs −γR )−vT ∆t cos(µU Abˆs +γT )] d∆t, S s ηS X X DA Spq, p˜ ˜q (ν) = 2S s=1 ˆ b=s−1 Z 2π ˜ T cos(µDAsˆb +θT )−(q−˜ q )dR cos(µDAsˆb −θR )] ej λ [(p−p)d

(C.2) ∞ −∞

½ · · ¸2 σ2 ˆ 2π 2π 2 exp − DAsb ∆tζ∆Z sin µDAsˆb cos µDAsˆb + DAsˆ b 2 λ λ 2π (q − q˜)dR sin(µDAsˆb − θR ) + vR ∆t sin(µDAsˆb − γR ) − λ ¸ ¾ £ ¤ 2 (p − p˜)dT sin(µDAsˆb + θT ) − vT ∆t sin(µDAsˆb + γT ) e

−j2πν∆t−

2 ∆tζ∆Z 2

DAsˆ b

[10] C. Bjerrum-Niese, L. Bjorno, M. A. Pinto, and B. Quellec, “A simulation tool for high data-rate acoustic communication in a shallow-water, timevarying channel,” IEEE Journal of Oceanic Eng., vol. 21, pp. 143–149, Apr. 1996. [11] M. Chitre, “A high-frequency warm shallow water acoustic communications channel model and measurements,” J. Acoust. Soc. Am., vol. 122, pp. 2580–2586, Nov. 2007. [12] A. Abdi and H. Guo, “Signal correlation modeling in acoustic vector sensor arrays,” IEEE Transactions on Signal Processing, vol. 57, pp. 892–903, Mar. 2009. [13] T. C. Yang, “Toward continuous underwater acoustic communications,”Proc. OCEANS’08, Quebec City, Canada, Sept. 2008. [14] T. C. Yang, “A study of spatial processing gain in underwater acoustic communications,” IEEE Journal of Oceanic Eng., vol. 32, pp. 689–709, July 2007. [15] L. M. Brekhovskikh and Y. Lysanov, Fundamentals of ocean acoustics 2e. Springer, Berlin, 1991. [16] W. C. Jakes, Microwave Mobile Communications, 2nd ed. Piscataway, NJ: Wiley-IEEE Press, 1994. [17] G. L. St¨uber, Principles of mobile communication 2e. Kluwer, 2001. [18] R. Kattenbach, “Statistical modeling of small-scale fading in directional radio channels,” Journal on Selected Area in Communications, vol. 20, pp. 584–592, April 2002. [19] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products . San Diego, CA, 5th ed., 1994. [20] L. J. Greenstein, D. G. Michelson, and V. Erceg, “Moment-method estimation of the Ricean K-factor,” IEEE Commun. Letters, vol. 3, pp. 175–176, June 1999. [21] A. G. Zaji´c, G. L. St¨uber, T. G. Pratt, and S. Nguyen, “Wideband mimo mobile-to-mobile channels: geometry-based statistical modeling with experimental verification,” IEEE Trans. Veh. Tech., vol. 58, pp. 517534, Feb. 2009. [22] R. Fletcher, Practical Methods of Optimization - Constrained Optimization 2e.. John Wiley and Sons, 1981.

2

[ 2π b] λ sin µDAsˆ

[vR ∆t cos(µDAsˆb −γR )−vT ∆t cos(µDAsˆb +γT )] d∆t. R∞ √ 2 2 2 2 Using the equality −∞ e−p x ±qx dx = πeq /(4p ) /p [19, eq. 3.323-2] to solve the integrals in (C.1) and (C.2), we obtain the D-psds in (47) and (48).

e−j

2π λ

R EFERENCES [1] M. Stojanovic, “Recent advances in high-speed underwater acoustic communications,” IEEE Journal of Oceanic Eng., vol. 21, pp. 125–136, Apr. 1996. [2] B. J. Davis, P. T. Gough, B. R. Hunt, “Modeling surface multipath effects in synthetic aperture sonar,” IEEE Journal of Oceanic Eng., vol. 34, pp. 239–249, July 2009. [3] T. E. Giddings, J. J. Shirron, “A model for sonar interrogation of complex bottom and surface targets in shallow-water waveguides,” Journal of the Acoustical Society of America, vol. 123 pp. 2024–2034, Apr. 2008. [4] T. B. Aik, Q. S. Sen, and Z. Nan, “Characterization of multipath acoustic channels in very shallow waters for communications,” Proc. OCEANS’06 Asia-Pacific, Singapore, May 2007. [5] M. Badiey, Y. Mu, J. A. Simmen, and S. E. Forsythe, “Signal variability in shallow-water sound channels,” IEEE Journal of Oceanic Eng., vol. 25, pp. 492–500, Oct. 2000. [6] D. B. Kilfoyle and A. B. Baggeroer, “The state of the art in underwater acoustic telemetry,” IEEE Journal of Oceanic Eng., vol. 25, pp. 4–27, Jan. 2000. [7] R. H. Owen, B. V. Smith, and R. F. W. Coates, “An experimental study of rough surface scattering and its effects on communication coherence,” Proc. OCEANS’94, vol. 3, pp. 483–488, Brest, France, Sept. 1994. [8] A. Essebbar, G. Loubet, and F. Vial, “Underwater acoustic channel simulations for communication,” Proc. OCEANS’94, vol. 3, pp. 495– 500, Brest, France, Sept. 1994. [9] A. Zielinski, Y-H. Yoon, and L. Wu, “Performance analysis of digital acoustic communication in a shallow water channel,” IEEE Journal of Oceanic Eng., vol. 20, pp. 293–299, Oct. 1995.

Alenka G. Zaji´c (S’99-M’08) received her B.Sc. and M.Sc. degrees from the School of Electrical Engineering, University of Belgrade, in 2001 and 2003, respectively. She received her Ph.D. degree in Electrical and Computer Engineering from the Georgia Institute of Technology in 2008. In 2010, she joined the School of Computer Science, Georgia Institute of Technology, where is visiting assistant professor. Her research interests are in wireless communications and applied electromagnetics. Dr. Zaji´c received the Best Paper Award at ICT 2008, the Best Student Paper Award at WCNC 2007, and was also the recipient of the Dan Noble Fellowship in 2004, awarded by Motorola Inc. and IEEE Vehicular Technology Society for quality impact in the area of vehicular technology.