System function and characteristic quantities of spatial ... - IEEE Xplore

System Function and Characteristic Quantities of Spatial Deterministic Gaussian Uncorrelated Scattering Processes Matthias Patzold Department of Information and Communication Technology Faculty of Engineering and Science, Agder University College Grooseveien 36, N-4876 Grimstad, Norway E-mail: [email protected]

Abstract - Spatial deterministic uncorrelated scattering (SDGUS) processes have been introduced recently to model and t o simulate space-selective wideband mobile radio channels. This paper studies t h e system functions and characteristic quantities of SDGUS processes. Starting from the space-time-variant impulse response, closed-from expression will be derived for the delay-direction-Doppler power spectral density (PSD) by using time averages. From this function, several other three-dimensional (3-D) and tw-dimensional (2-D) system functions can be obtained in closed form by using t h e Fourier transform. The Fourier transform relationships between all s y s t e m functions are illustrated. I n addition, closed-form solutions are presented, e.g., for t h e delay PSD, t h e direction PSD, and t h e Doppler PSD, only t o name a few. Moreover, simple expressions in terms of t h e model parameters will he provided for various correlation functions (CFs) and t h e main characteristic quantities of SDGUS models, such as the delay spread, the direction spread, as well as t h e Doppler spread. T h e obtained results provide a clear insight into the statistical behaviour of SDGUS models.

a

recently in [3] to enable the modelling and simulation of spaceselective wideband mobile radio channels. Following the ideas i n 111, the characterization of SDGUS processes is carried out in terms of system functions. Here, system functions are not only restricted to functions enabling to establish a relationship between the input and the output signal of a radio channel. In fact, under system functions, we will understand all those furictions, which give insight into various aspects of the behaviour of the radio channel. The paper is organized as follows. Section I1 reviews briefly the SDGUS model. For this model, 2-D and 3-D system functions are studied in Section 111. Starting from the space-time-variant impulse response, a closed-form sw lution will be derived in Section IV for the delay-directionDoppler PSD. As we will see in Section V, the delaydirection-Doppler PSD is of central importance, since it provides the basis for the computation of many other system functions, such as the delay-Doppler PSD, the direction-Doppler PSD, and the delay-direction PSD. Section V I studies some important characteristic quantities, e.g., the delay spread, the direction spread, and the Doppler spread. Finally, Section VI1 concludes the paper. 11. THESDGUS MODEL

I. INTRODUCTION Both, from the scientific and the technical point of view, it is desirable to develop a systematic channel characterization procedure enabling a deep insight into the physical propagation mechanism. Bello's seminal work [l] presents such a procedure for the characterization of randomly time-variant linear channels. He specialized his results by considering the class of widosense stationary uncorrelated scattering (WSSUS) processes, which are playing a major role in the modelling of wideband mobile radio channels till this day. In 121, Bello's system characterization is extended hy including direction dispersion and space selectivity in the radio channel. This paper is concerned with the characterization of SDGUS processes. Such processes have been introduced

0.7803-7757-51031$17.~ 0 2 W 3 IEEE.

In the following, the SDGUS mode1 introduced in [3] is briefly reviewed. We assume that a linear antenna array is mounted horizontally on the z-axis at the base station, whereas the mobile station is equipped with an omnidirectional antenna. Furthermore, we assume that the base station is the receiver and the mobile station is the transmitter. The complex baseband representation of the space-time-variant impulse response h(r',x,t ) of the SDGUS model at location x can be represented by a superposition of C propagation paths as follows

where

256

is a real-valued constant called the path gain of

the Ith propagation path, and 7; 2 0 is the corresponding path delay. The function pr(z,1 ) in (1) represents a spatio temporal complex deterministic Gaussian process

O,l,

..., L - 1 111. 2-D A N D 3-D SYSTEM FUNCTIONS

NI

&(.,t)

c ~ ,.j(Z,n,,,~=+z=f".,t+~",~) f

=

(2)

n=1

where P = 0, 1, . . . , C - 1. In ( 2 ) , Ne denotes the number of exponential functions of the Cth path, en,! is called the Doppler coefficient of the nth component of the Cth are the associated discrete Doppler path, and f,,e and frequency and phase, respectively. The quantity R,,c is called the (normalized) incidence direction, which is related to the azimuth angle of arrival &,,e and the waveIn the SDGUS model, .length Xa via R,,e = X;'sind,,e. all parameters (C,Ne, Ze, ?,; cn,e, On,!, fn,e, and &,,e) are ,per definition constant quantities, which have to be determined properly, e.g., by using the method described in [3] or by applying the perfect channel modelling approach 141. A perfect spati-temporal channel simulator enables the emulation of measured space-selective wideband m o bile radio channels without producing any model error or making approximations. Such a simulator is of central importance for the test and the performance investigation of future mobile communication systems with smart antennas under real-world propagation conditions. : t , A special case is obtained by considering the spacetimevariant impulse response h(r',x ,t ) a t location x = 0. Then, (1) reduces to the impulse response of deterministic Gaussian uncorrelated scattering (DGUS) processes [5] enabling the modelling of frequency-selective mobile radio channels. 'It should he 'mentioned that the DGUS model can be interpreted as the deterministic counterpart of the WSSUS model introduced by Bello [l]. Another special case is given wken z = 0 and C = 1. In this case, the impulse response h(r',x,t ) reduces to the impulse response of a flat fading channel, where the fading behaviour is modelled by using techniques basing on Rice's sum of sinusoids 15, 6 , 7, 81. In general, it is assumed that scattered components with different path delays are statistically uncorrelated. To impose this swcalled uncorrelated scattering (US) condition on the SDGUS model, requires that the complex deterministic Gaussian processes ( 2 ) are mutually uncorrelated. Hence, b e ( z , t ) and FA(., t ) have to he uncorrelated for e # A, where e, X = 0, 1, . . . , C - 1. It can be shown, however, that the US condition is satisfied if the sets of discrete Doppler frequencies { fn,e} and { fm,A} are mutually disjunct for different propagation paths. Thus, the US condition can be phrased as follows: US US

@

= 0 for

In the following, we assume that the SDGUS model satisfies the US condition (3). Furthermore, it is assumed that the discrete Doppler frequencies jn,e are chosen in such a way that f , L , f # jm,c for n # m and P = 0, 1, . . . , L - 1. Taking these conditions into account, we study the correlation properties of the deterministic Gaussian process by substituting (2) into

The solution of the above equation results in i p r p A ( A z , A t=) 0 for P # A, and for P = A, we obtain the 2-D system function N,

i,,,,,(Az, A t ) =

1 c:,~ eJZn(**fAz+f=

(5)

"-1

which is called the space-time CF. From this function, three other 2-D system functions can be derived in closed form by applying the Fourier transform, as illustrated in Fig. 1. For example, the time CFIdirection PSD R,,,,(R, At) is obtained by computing the Fourier transform of the space-time CF T l l e r C ( A t~), with respect to Ax. This is symbolically expressed in Fig. 1 by

fir,,, (n,At)

-

!2 A z .

T , ~ , ~ ( AW. X, Space-time CF

Direction-Doppler PSD Fig. 1.

Relationships between 2-D system functions of SDGUS models.

B. 3 - 0 System Functiom

&(x,t)andfix(x,t)are uncorrelated fore # X

Ifn,!) n { f m , d

A . 2-D System Functzons

e#X

(3)

where n = 1,.2 , . . . , N j > m = 1, 2, . . . ,N A , and P, X =

257

The space-time-variant transfer function H ( f ' , x,t ) of the SDGUS model is a 3-D system function, which is defined as the Fourier transform of the space-time variant impulse response & ( T I , x,t ) with respect to r', i.e.,

C-1

il:iptpt(Az, At) 6 ( ~ ;- Ti) 6 ( ~ ;- ?;)

=

(8)

e=o

Altogether seven 3-D system functions can he obtained in closed form from the space-time-variant impulse response h(r',z.t) by applying the Fourier transform. The relationships between them are illustrated in Fig. 2.

where FprPC(Az,At) denotes the space-time C F as inti-@ duced in ( 5 ) . From the fact that 6('r; - ?); 6(r; - 7;) is equivalent to 6 ( ~ ; 7;)6(r; - T ; ) it follows that (8) can be represented by ~

fhh(T;,Ti;Z,Z

+ AS; t , t t At)

= 6(T; - T i ) S h h ( T ; ,

AZ, At)

(9)

where C-1

Shh(T',Az,At)= ~ & ~ T p , p l ( A z , A t- )76; )( ~ '(10)

e=o is called the space-time CF/delay PSI). The 2-D Fourier transform of ihh(T',Az, At) with respect to Ax and At defines the delay-direction-Doppler PSD (scattering function) r - l . . .N. I

I

,!?(T',

Fig. 2 .

Relationships between 3-D system functions of SDGUS models.

.Iv. DERIVATION OF

THE SCATTERING

FUNCTION

'

Starting from the spacetime-variant impulse response, we will derive in the following a closed-from expression for the scattering function, which is also called the delaydirection-Doppler PSD. As mentioned above, all parameters of the SDGUS model are constant quantities. Consequently, the spacetimevariant impulse response h ( ~ ' , zt ,) is completely deterministic For that reason, the correlation properties of h ( ~ z, ' , t ) have to be derived by using time averages instead of statistical averages. Therefore, we define

R, f)

=

?i)

( & ~ c , , p6) ~( ~ -' e=o n = i . J(R - %,e) S(f - fn.e) .

(11)

This closed-form solution enables the analytical investigation of the delay-direction-Doppler characteristics of SDGUS models if the model parameters L, { N e } , { & e } , {cn,e}, {n,,,e}, and {fn,e} are known. The delay-directionDoppler PSD is of central importance, since this function provides the basis for the computation of many other system functions. By applying Fourier transform techniques, closed-form expressions can he derived for all the system functions shown in Fig. 3.

{Ti},

Space-time CFI

Fr4"e"CY-

delay PSD

space-rim CF

fhh(~;.r;;z,l:+A2;t,t+At) T

:.~ . lim

T-;o

h1(7;,z,t)j1(T;,z+Az,t+At)dt. (7)

-

2T

1

-Ti

Substituting the space-time-variant impulse response & ( T ' , z ,t ) [see (l)]in (7) and taking the US condition (3) into account gives

fhh(~;,~;;;z,l:+A~;t,t+At) c-1c-1

= lim T-CC

&?&A

e=o "

6 ( 4 - 7;) 6 ( ~ ;- ?);

Frequency CFI

Doppler PSD

Direction-Doppler PSD

Fig. 3.

A=O

.L jp;(z.t)jih("+Az,l+At)dt 2T -T

Dclay4ireCtiO"-

Fourier transform relationships between system functions of SDGUS models.

It can be seen by studying Fig. 4 that the delaydirection-Doppler PSD is also the starting point for the

258

derivation of several other important system functions, e.g., the delay-Doppler PSD, the direction-Doppler PSD, and the delay-direction PSD. From these system functions, the Doppler PSD, the delay PSD (also known as delay p r e file), and the direction PSD can he ohtained. Especially the last-mentioned system functions can he used to derive many important characteristic quantit,ies, such as the Doppler spread, the delay spread, and the direction spread, only to name a few. In the following, closed-form expressions will be derived for all those system fiinctions.and characteristic quantities which are presented in Fig. 4.

This result shows that the delay-Doppler PSD $ / ( T ’ , f ) is a finite double sum of weighted delta functions located in the (r’,f)-plane at positions (?j,f”,e), where the corresponding weighting factor is given by the constant

(6c,L,e)z. Direction-Doppler PSD: The direction-Doppler PSD f) is obtained by integrating the scattering function S(T’,12, f ) over the path delays T’ 2 0, i.e.,

&,(12,

m

Sn,(n,f) = /S(TJ.n,f)dT’

Dclay-di~crion-DopplcrPSD

\\

Mean

Doppler spread

Doppler

Mcan delay

Delay verd

Again, by- making use of the expression for the scattering function S(T,$, f ) [see (ll)],a closed-form solution can be ohtained for Snr(f,12). Hence,

From the result in (151, we conclude that the directionDoppler PSD Snf(Q, f ) is a finite double sum of weighted delta functions located in the (12,/)-plane at positions (Cln,e, f,ie), where the corresponding weighting factor is again given by (arc,,()’.

Mran Direction direction spread

shin

Fig. 4.

System functions and characteristic quantities of SDGUS

models.

v. PSDS AND

(14)

0

Delay.Direction p s D ; The delay.direction PSD-S,,n(T‘, 0) is related to the scattering function S(.r’, 12, f ) via

CFS

A . 2 - 0 PSDs of SDGUS Models In this subsection, we use the scattering function a starting point to derive closed-form solutions for the delay-Doppler PSD, the direction-Doppler PSD, as well as the delay-direction PSD of SDGUS models.

Substituting (11) into (16) results in the following c+d. form

Delay-Doppler PSD: The delayDoppler PSD $ f ( ~ ’ , f ) is obtained by integrating the scattering function S(T’, 12, f ) over all incidence directions 12, i.e., CC

(12)

The delay-Doppler PSD .!?+j(~’,f) is a 2-D system function, which describes the average power of the scattering components with path delay 7’and gives insight, into the phenomena caused by the Doppler effect. Using (11) allows us to present the delay-Doppler PSD S+,(T’, f) of the SDGUS model in a closed’form according to

c

C-1 Ne

S + ~ ( T 12) ’, = e=o

“=I

(Gec,,e)’6(~‘- ?i)d(n - 1 2 ” ~ ) .(17)

B. 1-D PSDs of SDGUS Models In this subsection. closed-form exoressions are or& sented for the delay PSD, the direction PSD, and the Doppler PSD of SDGUS models.

Delay PDS: The delay PSD using (13) as follows %(r’) =

7

S+f(T’,f)df

-m

S+(T’)

can be obtained by

C-I

5: S(T’ -

=

‘$1.

’

(18)

e o

where we have assumed without, restriction of generality

C-1 Ne

c:,~ 1 for all t = 0; 1, .. . , L - 1. cC(.ecn.e)’b(~’--i;)b(f~f,,e). (13) that e=o =

S+f(~’,f)=

n=i

259

Direction PSD: The direction PSD &(a) can he determined from the delay-direction PSD S,,n(T’, n) [see (17)] as follows m

x$$’(n) e=o

c-I

Sn(n);JS~,n(~~,n)di’= 0

(19)

where

where

refers to the delay-direction PSD of t,he lth propagation path. Doppler PSD: The Doppler PSD S,(f) can he derived from the delay-Doppler PSD S+,(T’, f ) [see (13)] as follows

refers to the space C F of the lth propagation path. Equation (26) shows that the behaviour of “)(A%) is completely determined by the model parameters Ne, cn,e, and Q,e,.

Time CF: The time CF ;,(At) is defined as the inverse Fourier transform of the Doppler PSD S,( f). Hence, from @I),it follows c-I

?,(At) =

$?:()(At)

(27)

e=o

where

where

.

7l=l

,

denotes the Doppler PSD of the lth propagation path.

C. 1-D CFs of SDGUS Models In the following, closed-form expressions are presented for the frequency CF, the space CF, and the time CF. Here, these CFs are obtained by invoking the WienerKhinchine theorem, which states that the C F and the PSD form a Fourier transform pair. Frequency CF: The frequency C F ?,,(AY)is defiFed as the inverse Fourier transform of the delay PSD S+(T’), I.e.,

1 cc

F+(Af’) :=

S+(T’) ej2”Af’r‘dT’,

designates the time CF of the lth propagation path. The result in (28) demonstrates that ?:(’(At)is completely determined by the model parameters Ne, cn,e, and fn,e.

VI. CHARACTERISTIC QUANTITIES

In this section, closed-form expressions are presented for some important characteristic quantities of SDGUS models, such as the delay spread, direction spread, and Doppler spread. Mean delay: The mean delay D$) of the SDGUS model is defined as the first moment (center of gravity) of the delay PSD S+(T’). Thus, by using (18), we obtain

(23)

0

Substituting the right-hand side of (18) in (23) gives the frequency CF i.+(Af’) in the closed form

Note that, for any given number C of propagation paths, the behaviour of F+(Af’) is completely determined by the model parameters ?it and i;

DZ)

of the SDGUS model Delay spread: The delay spread root of the second central moment is defined as the sq!are of the delay PSD S+(T’), i.e.,

Space CFr The space C F +*(A.) is defin_ed as the inverse Fourier transform of the direction PSD Sn(n). The computation of the inverse Fourier transform of (19) results in

D$)

:= 0

260

Mean dzrectzon: The mean direction B i ) is defined as the first moment of the direction PSD S n ( R ) . Thus, by using (19) and (20). we obtain

s" R Sn(n) d o

B p := -m

T.Sn(R)dR -m

L-I N I

C

- e=o -

(Ze c,,?)'

%,e (31)

C-1 N r

E E (62 c,,.e)' k 0 n=1

Direction spread: The direction spread Dg)is defined as the square root of the second central moment of the direction PSD Sn(R)_i.e.,

VII. CONCLUSION In this paper, the most important system functions and characteristic quantities giving insight into the statistical behaviour of SDGUS processes have been studied. The spacetimevariant impulse response was the starting point for the derivation of several 3-D system functions by using time averages and Fourier transform techniques. The relationships between all 3-D system functions have been established. A closed-form solution has been derived for the 3-D scattering function. Closed-form expressions have also been derived for 2-D PSDs (delay-Doppler PSD, directionDoppler PSD, and delay-direction PSD) as well as for 1-D PSDs (delay PSD, direction PSD, and Doppler PSD). Moreover, closed-form expressions have been provided for the main characteristic quantities of SDGUS models, such as the delay spread, the direction spread, and the Doppler spread. The obtained results provide a deep insight into the statistical behaviour of SDGUS models and give access to an analytical investigation of the system behaviour in terms of the model parameters.

REFERENCES In a similar way, the mean direction and the direction spread of the lth propagation path are defined.

Mean Doppler shift: The mean Doppler shift D y ) is defined as the first moment of the Doppler PSD f). Thus, by using (21) and (ZZ),we obtain

s,(

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Sons, 2002.

ay)

Doppler spread: The Doppler spread is defined as the square root of the second central moment of i.e.,

s,(f),

161 S. 0.Rice, "Mathematical analysis of random noise," Bell Syst. Tech. J . , vol. 23, pp. 282-332, Jul. 1944. 171 S . 0. Rice, "Mathematical analysis of random noise," Bell Syst.

Tech. J . , vol. 24, pp. 4G156, Jan. 1945.

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181 M. Pitzold and U. Killat and

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