Wireless Corner

~I

Nattali (Tuli) Herscovici

Christos Christodoulou

} .Lincoln Laboratory - Group 61 .r: Massachusetts Institute of Technology 244 Wood Street ~ Lexington, MA02420-9108 USA Tel: +1 (781) 981-0801 Fax: +1 (928)832-4025 Skype/AOL: tuli01 E-mail: [email protected]

Department of Electrical and Computer Engineering University of New Mexico Albuquerque, NM 87131-1356 USA Tel: +1 (505) 277 6580 Fax: +1 (505) 277 1439 E-mail: [email protected]

A Geometrical Model for the TOA Distribution of Uplink/Downlink Multipaths, Assuming Scatterers with a Conical Spatial Density Vue Ivan Wu and Kainam Thomas Wong Department of Electronic and Information Engineering, Hong Kong Polytechnic University Hung Hom, Kowloon, Hong Kong E-mail: [email protected], [email protected]

Abstract The time-of-arrival (TOA) distributions of the uplink and downlink multipath are analytically derived in this paper. This is based on geometrical models that simplify the spatial relationship among a mobile transceiver, the scatterers, and a base-station transceiver. These models idealize the scatterers as lying on a circular disc centered around the mobile transceiver, with these scatterers concentrating in a conically shaped spatial density. The base-station transceiver may lie either among these scatterers (in an indoor propagation environment) or outside this disc of scatterers (for an elevated base-station outdoor receiver). In contrast to the customary uniform-disc density, this "conical" scatterer density indirectly accounts for the multipath scattering power loss. These new TOA distribution formulas, herein derived explicitly in terms of the model's only two independent parameters, can better fit some empirical data than can all earlier models that also confine all scatterers to within a circular disc. Keywords: Communication channels; dispersive channels; fading channels; geometric modeling; microwave communication; mobile communication; multipath channels; scatter channels; land mobile radio propagation factors

196

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1. Introduction 1.1 The Propagation Delay in Wireless Communications

A

signal transmitted from a mobile user in a land mobile radiowave wireless cellular communication system arrives at the cellular base station through multiple propagation paths ("multipaths"). Each multipath carries its own propagation history of electromagnetic reflections and diffractions, and corruption by multiplicative noise: a history reflected in that multipath's amplitude, Doppler, arrival angle, and arrival time delay at the receiving antenna(s). The values of these amplitudes, Doppler frequency shifts, arrival angles, and arrival time delays depend on the electromagnetic properties of and the spatial geometry among the mobile transmitter, the scatterers, and the receiving antennas. Each receiving antenna's data measurement sums these individually unobservable multipaths. A channel's impulse response (IR) may be represented by a linear time-invariant filter, IT (T), if the channel is (or can be approximated as) temporally stationary. The corresponding spectrum would generally not have a flat magnitude over the frequency coordinate: hence the term, "frequency selective." Only in the degenerate case of IT (T) being a single impulse would the channel's spectrum have a flat magnitude response over all frequencies. T, produces frequency distortions (via a Fourier-type transformation). These are to be distinguished from spreading in the frequency coordinate, which corresponds to temporal variability in the channel (i.e., temporal non-stationarity in the channel's impulse response).

In other words, spreading along the delay coordinate,

as "bad urban" city blocks, with high-rise buildings as scatterers on all sides, or "rural" settings, with few scatterers close by an elevated base station. One geometrical model can apply to a wide class of propagation settings, producing the received signal's measurable fading metrics, generally applicable within that class of channels. "Geometrical modeling" achieves such generalization by idealizing the complex wireless physical environment via geometrical abstractions of the spatial relationships among the transmitter (Tx), the scatterers, and the receiver (Rx). While the present metric of interest is the propagation delay, the underlying mechanism that characterizes this metric involves the spatial relationships among the transmitter, the scatterers, and the receiver. A multipath's delay reflects the spatial distance traversed by the signal on the multipath. Hence, the delay profile is intricately related to and founded on the spatial relationships. Geometrical models attempt to embed measurable fading metrics (e.g., the uplink multipaths' TOA distribution) integrally into the propagation channel's idealized geometry, such that only a lew geometrical parameters would affect these various fading metrics in an inter-connected manner, to conceptually reveal the channel's underlying fading dynamics for a wide class of physical realities. Geometrical modeling also contrasts with an ad hoc non-geometrical model that imposes certain ad hoc and a priori statistics on each diverse aspect of the multipaths' spatial and temporal behavior, without embedding these presumptions into an integrated comprehensive geometrical model. Because no underlying geometrical interconnection exists in such a non-geometrical model, such nongeometrical models (perhaps useful as curve-fitting devices) reveal little analytical insight into the propagation channel's fundamental dynamics, and offer no conceptual framework to facilitate meaningful generalization to different propagation settings.

For each uplink multipath over which a signal travels from the mobile transceiver to the base-station transceiver, there can exist a corresponding downlink multipath, traversing the same spatial path but in the opposite direction, from the base-station transceiver to the mobile transceiver. Hence, the uplink IT (T) equals the downlink

IT (T ) •

The time-of-arrival (TOA) probability-density function characterizes the wireless-propagation channel's temporal delay spread and frequency incoherence. These, in tum, determine the obtainable temporal diversity, and the extent of inter-symbol interference, in wireless communication. These constrain the capacity of information that can be communicated between the transmitter and the receiver. Incidentally, many applications (such as singleinput single-output communication systems) are interested only in the above-mentioned temporal metric, but not in any spatial metric.

1.2 Geometrical Modeling This TOA distribution can be measured (or computer estimated) in site-specific or terrain-specific or building-specific empirical measurements (or in ray-shooting and ray-tracing computer simulations). However, such results would be applicable only to the particular propagation setting under investigation, and cannot be easily generalized to a wider class of scenarios. A rough model, applicable to a wide class of field scenarios, could be useful to the system-development engineer to develop his/her products. The products must be usable in a wide class of environments, such

1.3 Survey of Geometrical Models to Derive TOA Distribution Within the geometrical modeling literature that analytically derives closed-form formulas for the TOA distribution explicitly in terms of the model parameters, one of the simplest and most common class of geometrical models is the class of "circular-disc" models. In these, all scatterers are idealized to be spatially distributed only within a circular disc, according to different spatial densities in different geometrical models. This circular disc centers upon the mobile transceiver, whereas the base-station transceiver may lie either inside or outside the disc. Within this class of geometrical models: 1.

The scatterers are modeled as uniformly distributed within the circular disc, 1a.

with the base-station transceiver lying outside the circular disc [1]. This model can apply where an outdoor base-station transceiver is placed on an elevated tower, and thus has few scatterers in the base-station's immediate vicinity. Please see Figure 1a.

1b.

with the base-station transceiver lying within the circular disc [2]. This model can apply for "bad urban" or indoor scenarios, where the base-station transceiver is surrounded by scatterers. Please see Figure lb.

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197

5.

Figure la. The scatterers' spatial support region in the proposed outdoor model with the base-station (B8) transceiver on an elevated tower and away from the dominant scatterers.

4a.

focused at the mobile transceiver and the basestation transceiver [1, 5, 6],

4b.

centered at the mobile transceiver alone, with the base-station transceiver outside of the ellipse [6, 7],

A three-dimensional hemi-spheroidal region above the mobile transceiver [8].

Unlike the abovementioned uniform-density models in Models 1a and 1b, the "conical" spatial density (like the inverted-parabolic-shaped density of Model 2) can account for the more-frequent reflections off scatterers nearer to the mobile transceiver. In the above geometrical abstraction, a reflection farther from the mobile transceiver intuitively may correspond in physical reality to a sequence of consecutive reflections occurring spatially farther and farther away from mobile, but these reflections incur power loss. The consequence is roughly equivalent to more single-bounce scatterers closer to the mobile transceiver. Rather than modeling the scatterers' re-transmission power as spatially non-stationary, it is mathematically simpler to model scatterers to have identical retransmission characteristics, but more densely spaced the closer to the mobile transceiver. Instead of assuming lossy scatterers (which would further complicate the present mathematical derivation), the present model has a lower spatial density of scatterers where the physical propagation paths would likely have their "last bounces" (and would have already suffered much reflection power loss) before reaching the base-station transceiver. The scatterers' conical-distribution geometrical model was in fact first proposed in [9], but only the azimuth direction-of-arrival (DOA) distribution was derived and only for the uplink. No TOAdistribution was derived in [9]. This paper will fill this literature gap.

2. The Presently Advanced "Geometrical" Models Figure 1b. The scatterers' spatial support region in the proposed indoor model, with the base-station (B8) transceiver lying among the scatterers.

2. The scatterers are modeled as distributed on the circular disc according to an inverted-parabolic-shaped density, with the mobile transceiver lying outside the circular disc for a base-station transceiver on an outdoor elevated tower, and thus with few scatterers in the receiver's immediate vicinity [3]. Please see Figure 1a. (The open literature currently has no result for the case of the scatterers being distributed on the circular disc according to an invertedparabolic-shaped density, with the mobile transceiver lying inside the circular disc.) The above circular-disc support region has been generalized

Figures 1a and 1b show the spatial geometries relating the mobile transceiver, a scatterer, and the base-station transceiver. Let the base-station transceiver (BS) be located at the origin of a twodimensional plane, whereas the mobile transceiver (MS) is located at the Cartesian coordinates (D, 0), with D being the base stationmobile transceiver distance. Let the aforementioned circular disc's radius be R. The scatterers' spatial locations are idealized as conically distributed within this circular disc [9]:

0,

otherwise

(1) Figure 2 illustrates this conical spatial density.

3.

A hollow circular disc (on which the scatterers are uniformly distributed) [4];

For an outdoor base station on an elevated tower (and thus away from any dominant scatterer), the D ~ R case in Figure la applies. For an indoor or "bad urban" base-station transceiver lying among from the scatterers, Figure 1b's D ~ R case applies.

4.

An elliptical disc (on which the scatterers are uniformly distributed)

The propagation time-of-arrival (TOA) equals r = ro + rs for c

to:

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(3)

~ ..........

i···· ""

I

.... ~ ...

Next, the bivariate spatial density is transformed into a bivariate density of the azimuth direction-of-arrival (DOA) and the time-ofarrival (TOA). This is achieved through a Jacobian transformation. Where the constraint in Equation (2) is satisfied [9, 1],

..... :

." -.. ~.... "" ....

10~

'::~".,:: ... 500

2000

o

where

·500

Y. in meters

·1000

0

x. in meters

Figure 2. The conical spatial density, at R = 500 m and D=1000m.

I~:I

r

c[(~ -2~COSO+l]

(4)

2(COSO- ~r

a propagation path from the mobile transmitter, reflecting off a scatterer at (x, y), and arriving at the base-station transceiver. Here, c denotes the speed of propagation, T is the propagation time of arrival, B refers to the azimuth angle of that scatterer as seen by

the receiver, Ys

= J( x - D)2 + y2

,and

Yo

denotes the distance

between the base-station transceiver and any scatterer, as shown in Figure la. Like all earlier papers that analytically derive closed-form explicit expressions of the TOA distribution based on geometrical models, these following four standard assumptions are made: 1.

Each propagation path, from/to the mobile transceiver to/from the base-station transceiver, reflects off exactly one scatterer.

2.

Each scatterer acts (independently of other scatterers) as an omnidirectional lossless re-transmitter.

3.

There are negligible complex-phase effects in the receiving antenna's vector summation of its arriving multipaths. That is, all arriving multipaths arriving at each receiving antenna are assumed to be temporally in phase among themselves.

4.

(5)

Hence,

1,,8 (T,B)

DC[l-(~r][(~r -2~COSO+l] 4(cosO-

~

f

3c 41rR

Polarization effects may be ignored.

3. The Conical Model's TOA-DOA Joint Distribution To derive the density distribution,

I, (T),

(6)

Where the constraint in Equation (2) is violated, of the times-of-

arrival, the above spatial density's circular-disc support region is first expressed in terms of the polar coordinates (re,B), giving [3] (2)

Applying the cosine law to the geometries in Figures la and Ib [1, 3],

1,,8 (T,B) = O.

The TOA distribution may then be obtained by integrating 1,,8 (T,B) with respect to 8, as will be done in the next section.

Note that the TOA must satisfy

TE[

~ , D:2R ].

This is

because no propagation path can traverse the mobile station-base D station distance D with under of propagation delay. On the c

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199

other hand, the latest-arriving propagation path must have bounced off a scatterer on the circle's circumference at a direction diametrically opposite the base-station-to-mobile-station direction. and

4. The Outdoor "Conical" Model's TOA Distribution

Xo =

In outdoor propagation environments where a base-station transceiver is housed on an elevated tower, few scatterers will surround the mobile transceiver. The scatterers may thus be modeled to cluster only around the mobile transceiver. This corresponds to the R

~

. FIgure . 'fi D case In 1a. For any specIIc

T E

[D D+2R] , ~,--c-

there exists a T -constant spatial ellipse focusing at the basestation's and the mobile-station' s spatial locations. Any propagation path must bounce off a scatterer lying on this ellipse's rim. This elliptical rim intersects with the circle (within which the scatterers lie) at two points at most, namely at Ye (00 ) == TC - R in Figure la, where

Mathematica's symbolic integration produced a closed fonn for

I1Con,Out) ( T)

that explicitly depends on the model parameters of

Rand D as follows: liCon,Out) (T)

2

h(0 - 2& + 1)( 70 -4& + 4&2 -4)

c

&2~82_1

8nR 2

6( 20 -1)(0 - 2&) arctan ( h)

&2~82

(7a)

-1

where 8= TC D' R D'

These considerations lead to an integration range of B E [ -Bo, Bo] . Hence, the TOA's marginal density equals and

I1 Con ,Out) (T) = leo flo Ii (} (T,B)dO '

2nR f),

TC

2

-

o ( D)

D

J (cosB- TCD )3

o

h=

2

TC -2-cosB+1

&=-

TC)2 TC ( D -2 D cosB+l 2

~(~ -COSO)

dB

1-8+2& 1+8-2&

The T -constant ellipse can intersect the circle at not more than two points, as shown in Figure 3. Otherwise, R would exceed the distance from the mobile to certain points on the elliptical rim, thereby contradicting the fact that the shortest distance from the mobile to the elliptical rim is via point a in Figure 3b. (7b)

To summarize, the "conical" model (for outdoor environments with an elevated base station) has a TOA density equal to

. f:Con,OUI)(T)= EquatIOn (7a), { 0,

where 8, f!Con,Out)

Jt

where the last equality above has used the following transformations [3]:

B x = tan-, 2

l-x 2 cosB=-1+x 2 ' 200

&,

TE

[D D+2R] ~,--c- ,

(7b)

Otherwise

and h have been previously defined. Note that

(~) " pe aks at

T

Out) (T) = -D ' . = T(Con O' . B y recIprocIty C

between the base station and the mobile for the propagation delay, the above-derived formula applies for the uplink as well as the downlink.

5. The Indoor "Conical" Model's TOA Distribution In indoor or "bad urban" environments, scatterers may be omnipresent, even in the base-station's immediate vicinity. This IEEE Antennas and Propagation Magazine, Vol. 50, No.6, December 2008

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iC)2 iC ( D -2 D cos(}+l 2 ~(~ -COSO) 3D 2c

d()

-1](2!iiC) [2(iC)2 D D D

~ 3c (28 2 -1)(2& - 8)

Figure 3a. This figure helps in obtaining the appropriate integration range of () . This result was used in deriving fiCon,Out)

(i).

For

(8a)

&2~82_1

8R

\-I~ v" E [2R -D, 2R +D] , c

c

the

i

-constant ellipse inter-

sects with the circle at exactly two points, just as in the preceding section. Here, the () integration range equals [-(}o, (}o]. Mathe-

matica's symbolic integration gave

os

a

D

B.

iC

iC

2

2JrR 2

iC)2 iC ( D -2 D cos(}+l 2 ~(~ -COSO)

-2-cos(}+1

-

) ( D o D

( cos()- iC )3 D

of

c

Figure 3b. This shows that the i -constant ellipse can intersect with the circle at only two or fewer points. This fact was used in deriving fiCon,Out) ( i ) •

corresponds to the R ~ D case in Figure 1b, where the base station lies among the circular disc of scatterers. For this indoor or "bad urban" model, the i -constant ellipse intersects with the circle (within which the scatterers lie) under three disjoint cases: For V r

E[

~ , 2R:D ),

the r -constant ellipse lies wholly

[ -Jr ,Jr ) .

Mathematica 's symbolic integration gave

h(0-2&+I)(70 2 -4&+4&2_ 4)·

&2~82_1 2 6( 20 -1)(0 - 2&) arctan (h)

8JrR

&2~82_1 (8b) For all other

i

values, fiCon,In) (i) = o.

To summarize, the "conical" indoor model's (or the "conical" "bad urban" model's) TOA density equals

within the circular disc. Hence, to obtain the TOA marginal density, fr,B (i,(}) , in Equation (6) is to be integrated over

BE

d()

Equation (8a),

i E

.

fiCon,ln) (i)

Equation (8b), r

[D ,2R - D) C

E

C

[2R:D,2R:D] (8c).

0, Otherwise

To obtain the fiCon,In) (i)

peaks,

i

value (labeled a

solution

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iaCon,In»)

to

the

at which equation

201

d [ficon,In)

(i)J

=0

'

di This gives

. wIth

d 2 [flcon,In) (T)

D {~+ ~3Z2

i(Con,In) =

o

c

4

12

di 2

+.J6 12

18&2 + 21

-----+

(i;

J< 0

where '

is obtained.

[3&2 + 12 _ 3/Z 'ILl

3.J?,&3 -18.J?,& ]112 )

JZ;

,

(9)

where and

and Note that f;Uni,Out) ( i) peaks at i = iaUni,Out) ( i ) = D . 2

C

2 12 23f7Z 36& +42 Z 2 -3 - & + + VLI + 1f7

~ZI

By reciprocity between the base station and the mobile for the propagation delay, the above-derived formulas for fiCon,In) (i) and for iaCon,In) apply for the uplink as well as for the downlink.

6. Comparing The "Conical Circular Disc" Models to Earlier "Circular Disc" Models

6.2 Model 1b: The "Uniform Scatterer Density Circular Disc" Model for Indoors [2] If the base station lies inside the unifonn-density scatterer region, the TOA distribution of Equation (la) becomes

The above "conical-disc" models are proposed as close variants and alternatives to the customary "circular-disc" models of Equations (la), (lb), and (2) in Section 1.3, and further elaborated on below.

6.1 Model 1a: The "Uniform Scatterer Density Circular Disc" Model for Outdoors [1] For this customary model (with the base station lying outside the uniform-density scatterer region), the TOA distribution is given by

TE[~' 2R;D)

TE[2R;D, 2R:D]

(11)

Otherwise

0, where

So = arccos

1+ 28& _8 2

2&

,

fiUni,Out) ( i ) =

2 2 8 kok4 + 8kOk]2 -C- {1r8 k 2 - 8ki + 1rk2kl + 8kr - 2&kr + ----'----'------';.........;;...

1rD&2

fluni,Out) ( T )

= {

202

2kl +2k~kl

4kl k2

. EquatIOn (lOa),

TE

0,

Otherwise

[D 2R+D] -;;' - c - ,

(lOb)

J'

S2

= arctan(~c58 +-1 tan So 2

S5

= 8 -&,

S6

= c5 sin So + SfS5 cos So c5Sf sin So

1

2S4

4&S4 sin So

4&sl

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2500

...

~.:

' 'r .............. :.~

Note that fiUni,ln) ( i) peaks at is

because

i E [

i E [

i = iaUni,ln) ( i) = D(2e -1) . This increases

fiUni,In) ( T )

~ , 2R:D ),

c monotonously

but

decreases

for

monotonically

for

2000

.... .~.. ..... , ... ~ .....

. ..

. . . ..

... :.::

'

~

............. ~ . " ........

...... ~

..... ·0 ...

".

2

0 s=R/D

5='tc/D

Figure 4b. The TOA distribution of the customary (Inv-Out) "geometrical model" Model 2, which has the base station lying outside the circular disc, wherein the scatterers follow a inverted-parabolic spatial density.

....... ~

--_C_-(48et5 3GQ2 -18t5Q.j + 23GQ2 t5 4 96Q21tD(I- 8)c 2 5 2 -28c8QIQ2 + 192c 28 Q3 + 238 QIQ2 + 488 Q3 4 2 -36c 2QIQ2 -I28c 38QIQ2 -I92c 8 3Q3 + 96c QIQ2 -488 3QIQ2 -38a 30IQ2 + 188QIQ2 -488 Q3

f

~

~

.

~

···f··········..:··· ..

1500

iE[~, 2R;D]

0,

Otherwise

............. ~

2

...... .~

~.. .

-

~

:

.....

..~

.

'~""'"

:"

~

.

'

'

.

~

1500

..

.~ :;)

'

~

. '"

'

.

~

........

;

"~""" '

~~ 1(00

s= RID

Figure 4c. The TOA distribution of the (Con-Out) "geometrical model" advanced in this paper, which has the base station lying outside the circular disc, wherein the scatterers follow a conical spatial density.

..

.......... ~ ....... ... '~"'"

.. ;,

.

0

6='tc/D

f' .

~

........ ;....

o o

(12)

: .

..

500

2 2 2 2 +96e t5Q3 + 96e t5 GQ2 +40e t5GQ2)'

....... ~

........ ~

........ ~ ....

o

4

~

. ! .

o

2500

~

-' .

o o

This represents one nonuniform-density alternative to the outdoor model in Model 1a. Again, for a base station lying outside the circular disc (within which lie the scatters), but according to an inverted-parabolic spatial density, the TOA distribution is given by

f2COO

:

..

500

6.3 Model 2: The "Inverted Parabolic Scatterer Density Circular Disc" Model for Outdoors [3]

........:

.

..~ .. ,

1000

2R: D , 2R; D ] .

3000

,

"

'f>

';.....

.

........ ; ..

$" 1500 o ~-

":"

"

··..·r······("..·

~

,

where

Qo =8+1,

.. ~

.. ..~.. '. "

OI=

500

-1+28 -28-8 2 +288 1+ 28 - 2c + 8 2 - 28c '

o

o 2 5='tc/D

0

and s= RID

Figure 4a. The TOA distribution of the customary (Uni-Out) "geometrical model" Model 1a, which has the base station lying outside the circular disc, wherein the scatterers follow a uniform spatial density.

Q3 =arctan(QoQl/Q2)· The model variable c = R/D , which controls the shape of the model geometry, is common to the model fonnulas of Equations (7), (8), (10), (11), and (12). The model variable D, which

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203

scales the spatial size of the entire model geometry, is also common to these equations. Figures 4a through 5b compare all five TOA distributions, illustrating the following: 1.

The support range of five models.

2.

For each

value,

&

increases as

T

1.8

increases for all

&

• Empirical Data - - Proposed Conical Model - - - Customary Uniform Model - .- .. Customary Invert-Parabola Model

1.6

1.4 1.2

11·,Out) (T)

~

peaks at the initial value

.1:::

~

(I)

c::

of (} = 0, and monotonically decreases towards zero as (} increases. In other words, the first arriving non-lineof-sight (NLOS) propagation path is always the strongest propagation path. Moreover, the drop to zero for Con Uni ,Out) ( T) is more abrupt than for ,Out) ( T).

I1

...

~

G)

a ~ 0.8

....

~

...

0.6

I1

0.4

.~

\

...

....I ...~ ...

0.2

3.

For each

value, no

&

11·,ln) (T)

is monotonic. In other

.~.5

words, the first arriving non-line-of-sight propagation

.......j .

...... ~ ..... 150

~

~ ...

8

~~

o

•

~

';i

........... ~ .. '

c:: 'Q) 0

..~

o

g=

... \

...: 250

. ....~.....

.... " ........ ~ ........

o

50

o

1

o 4

RID

8

6 &='tc/D

Figure 5b. The TOA distribution of the (Con-In) "geometrical model" advanced in this paper, which has the base station lying inside the circular disc, wherein the scatterers follow a conical spatial density. 204

... 0.2

0.25

0.4

<

(Uni,In)

'0

.

~

~~ 100

g=

0.15

Figure 6b. The (Con, In) model better fitted this empirical dataset than the (Uni, In) model.

"0

..... ; .: .......

0.1

Relative Delay ('tre> , in fJ.S

-r(Con,In) .......:

..... ..........

.......:......

0.05

path is no longer always the strongest propagation path. Moreover, for any particular &, it is always true that

: .....

... : ......

o

...

.

... ., .... :.

S

...

~='tc/D

RIO

Figure 5a. The TOA distribution of the customary (Uni-In) "geometrical model" Model 1b, which has the base station lying inside the circular disc, wherein the scatterers follow a uniform spatial density.

E

..

Empirical Data - - Proposed Model - - - Customary Model

6

2

150

3

4

1

~!

2.5

~ f-

o

200

2

(\el. in ~

...

..... ~ ...

::3

50

1.5

Relative Delay

10 100

.

12 '"

!

~

£'

1

0.5

0

...

Figure 6a. The (Con, Out) model better fitted this empirical dataset than the (Uni, Out) and (Inv, Out) models.

r····

200

\'~

Figure 6a shows that the proposed (Con, Out) model (i.e., the base station lies outside the circular disc, wherein the scatterers follow a conical spatial density) better fits certain empirical data [10] taken in urban Chicago than can the customary (Uni, Out) and (Inv, Out) models. The above-mentioned field measurements involved an elevated base station atop a building that was on average 1~O ft above the surrounding terrain. The mobile receiver was in a vehicle and 2.7 m above ground. The channel-sounding signal was 20 MHz in bandwidth and centered around 3.676 GHz. The model calibration by the present authors was via minimization of the mean-squared error (MSE) between the "normalized" empirical dataset of the arrival-delay distribution {( 'j, Yj ), i = 1, ... , I} and the corresponding values {( Tj,f(X) ( Tj -T\ +

~ )). i = 1,... ,I}

from the geometric model's TOA distribution, IEEE Antennas and Propagation MagaZine, Vol. 50, No.6, December 2008

Authorized licensed use limited to: Hong Kong Polytechnic University. Downloaded on January 12, 2010 at 23:22 from IEEE Xplore. Restrictions apply.

(13)

where X

E

{(Con,Out),(Con,In),(Uni,Out),(Uni,In)}. Note that this

"normalization" was to ensure unit area under the empirical dataset. The normalization proceeded as follows: Let Yi denote the empirical value as presented in [11], and let ai,i+l denote the area of a trapezoid with its four comers at

('i' 0),

('i' Yi)' ('i+l' 0) ,

('i+l ,5'i+l ). The normalized empirical value at _ Yi -

Yi

I-I

'i

is then

.

Li=1 ai,i+l

The best-fitting (Uni, Out) model, at D = 707 m and R = 169.68m, suffered an MSE of 0.012782. The best-fitting (Inv, Out) model, at D = 326 m and R = 208.64 m, suffered an MSE of 0.010255. In contrast, the best-fitting (Con, Out) model, at D = 262 m and R = 222.7 m, suffered an MSE of only 0.009325, which was 27% less than that of the (Uni, Out) and 9% less than that of the (Inv, Out) models. Likewise, Figure 6b shows that the proposed (Con, In) model better fits certain in-building empirical data [12] than does the customary (Uni, In) model. The channel-sounding signal had a 250 MHz bandwidth in a seriously clustered environment. The best-fitting (Con, In) model, at D = 4m and R = 28 m, suffered an MSE of 1.2143, whereas the best-fitting (Uni, In) model, at D = 10 m and R = 25 m, suffered a 57% higher MSE of 1.9039.

7. Conclusion An idealized spatial geometry among the base-station transceiver, the scatterers, and the mobile transceiver has been presented above for indoor or outdoor wireless cellular communications. These scatterers are spatially confined to a circular disc centered around the mobile transceiver. The scatterers are distributed according to a conical spatial density, in contrast to the customary uniform density. The scatterers are modeled as omnidirectional lossless re-transmitters of incoming rays from the transmitter. Closed-form expressions for the uplink/downlink multipaths' timeof-arrival distributions were analytically derived.· These expressions are explicitly in terms of the two model parameters of the idealized geometry. This geometric model was shown to better fit certain empirical TOA data than the more-customary uniform-density or inverted-parabolic-density scatterer geometries.

8. Acknowledgment The work was supported by Internal Competitive Research Grant number G.42.R6.YF52 from the Hong Kong Polytechnic University.

9. References 1. R. B. Ertel and J. H. Reed, "Angle and Time of Arrival Statistics for Circular and Elliptical Scattering Models," IEEE Journal on Selected Areas in Communications, 17, 11, November 1999, pp. 1829-1840. 2. L. Jiang and S. Y. Tan, "Geometrically-Based Channel Model for Mobile Communication Systems," Microwave and Optical Technology Letters, 45, 6, June 2005, pp. 522-528. 3. A. Y. Olenko, K. T. Wong, and M. Abdulla, "Analytically Derived TOA-DOA Statistics of Uplink/Downlink WirelessCellular Multipaths Arisen from Scatterers with an Inverted-Parabolic Spatial Distribution Around the Mobile," IEEE Signal Processing Letters, 9, 7, July 2005, pp. 516-519. 4. A. Y. Olenko, K. T. Wong, and E. H.-O. Ng, "Analytically Derived TOA-DOA Statistics ofUplink/Downlink Wireless Multipaths Arisen from Scatterers on an Hollow-Disc Around the Mobile," IEEE Antennas and Wireless Propagation Letters, 2, 2003, pp. 345-348. 5. S. Y.-D. Lien and M. Cherniakov, "Analytical Approach for Multipath Delay Spread Power Distribution," IEEE Global Telecommunications Conference, 1998, pp. 3680-685. 6. M. T. Simsim, N. M. Khan, R. Ramer, and P. B. Rapajic, "Time of Arrival Statistics in Cellular Environments," IEEE Vehicular Technology Conference, 6, Spring, 2006, pp. 2666-2670. 7. M. R. Arias and B. Mandersson, "Time Domain Cluster PDF and Its Application in Geometry Based Statistical Channel Models," IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, 2007. 8. A. Y. Olenko, K. T. Wong, S. A. Qasmi, and J. AhmadiShokouh, "Analytically Derived Uplink/Downlink TOA and 2DDOA Distributions with Scatterers in a 3D Hemispheroid Surrounding the Mobile," IEEE Transactions on Antennas and Propagation, AP-54, 9, September 2006, pp. 2446-2454. 9. P. C. F. Eggers, "Generation of Base Station DOA Distribution by Jacobi Transformation of Scattering Areas," lEE Electronics Letters, 34, 1, January 8, 1998, pp. 24-26. 10. M. D. Batariere, T. K. Blankenship and J. F. Kepler, "Wideband MIMO Mobile Impulse Response Measurements at 3.7 GHz," IEEE Vehicular Technology Conference, 1, Spring 2002, pp.26-30. 11. S. Y. Seidel, T. S. Rappaport, S. Jain, M. L. Lord and R. Singh, "Path Loss, Scattering, and Multipath Delay Statistics in Four European Cities for Digital Cellular and Microcellular Radiotelephone," IEEE Transactions on Vehicular Technology, 40, 4, November 1991, pp. 721/~30. 12. P. J. Cullen, P. C. Fannin and A. Molina, "Wide-Band Measurement and Analysis Techniques for the Mobile Radio Channel," IEEE Transactions on Vehicular Technology, 42, 4, November 1993, pp. 589-603. @)

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