The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

1FULL CHANNEL CORRELATION MATRIX OF A TIME-VARIANT WIDEBAND SPATIAL CHANNEL MODEL

Hui Xiao Alister G. Burr Department of Electronics, University of York Department of Electronics, University of York York, UK York, UK

ABSTRACT

In this paper, the short-term time variation of wideband channels is investigated based on some extensions to the WINNER interim beyond-3G (IB3G) spatial channel model (SCM), using the full channel correlation matrix. These extensions mainly include some modifications of the generation of the wideband fast fading channel matrix, such as the consideration of the time variant sub-path phases at the mobile station (MS), and of the time variant path powers, and the generation of different last bounce distances (LBDs) for the mid-paths within each path. In order to measure the short-term time variation of the wideband channels, this paper presents a redefinition of the correlation matrix distance (CMD) metric, which was originally proposed for the narrowband fast fading scenario. By comparing the simulated CMD generated from the newly extended 3GPP (NE-3GPP) SCM with that of the measurements taken in the FLOWS project, we find the following main conclusions: (1) The Kronecker assumption can not be applied in calculating the instantaneous full channel correlation matrix, since it leads to significant deficiencies in the simulated CMDs from the NE-3GPP SCM. (2) The simulated CMDs generated from the NE-3GPP SCM compares better to the measurements taken in the FLOWS project than that of the IB3G SCM.

1 This work is partly supported by the Network of Excellence (NEWCOM).

I. INTRODUCTION

Multiple-input multiple-output (MIMO) has emerged as one of the most promising future technologies in mobile radio communications. It has been shown that the capacity of a MIMO system can scale linearly with the number of antennas in independent Rayleigh flat fading channels. However, in the real world, the multi-paths of a wideband MIMO wireless system have spatial correlation and time dispersion, which may significantly affect the MIMO performance. Further research in this field necessitates a more accurate MIMO channel model to investigate and evaluate new MIMO wireless systems under realistic channel conditions. In the European WINNER project, the popular 3GPP SCM for a MIMO wireless system is used to simulate the wideband fast fading scenario [6]. They assumed that the positions of the base station (BS) and scatterers are fixed during a short channel observation period in this wideband fast fading scenario. Thus the time variation of the channels is only due to the movement of the MS, which is a valid assumption in many cases of practical interest. Some extensions are made to

the 3GPP SCM to evaluate B3G technologies in outdoor scenarios within the WINNER project, such as: • the consideration of intra-path (or intra-cluster) delay-

spread, which is zero in the 3GPP SCM • the short-term time-variability of some channel

parameters, mainly including the time variant path delays, the time variant angles-of-arrival (AoAs) of the paths and the time variant shadow fading, which are taken as constant during a short time period in the 3GPP SCM

However, there is still more work to do to generate a wideband time variant channel matrix which is more comparable to the real situation, for example: • Although the intra-path delay spread is considered in the

IB3G SCM, the LBDs of the mid-paths (see section IIA) within a specific path, which is the distance between MS and the last bounce scatterer of each mid-path, are the same as the LBD of that path.

• Since the phases of the multi-paths that arrive at the MS are time variant, this should be taken into account when generating the instantaneous channel matrix, which was not done in the IB3G SCM.

• The time-variability of the multi-path powers should be considered when generating the instantaneous channel matrix, which was taken as constant in the IB3G SCM.

Therefore, following the extensions included in the IB3G SCM, we selected the same scenario to investigate the characteristics of the wideband fast fading MIMO channels, in which the time variation of the channels is only due to the movement of the MS. Our main contributions are: (1) generating the time variant AoAs at the MS, (2) generating different initial LBDs for the mid-paths, assuming a log-normal distribution, (3) when generating the instantaneous wideband channel matrix, the phases of the multi-paths at the MS are taken as time variant, (4) the time-variability of the multi-path powers is considered in the calculation of the instantaneous wideband channel matrix. In this paper, the instantaneous full channel correlation matrix at each frequency can be generated to describe the instantaneous spatial structure of the wideband channels. To analyze the time variation of the instantaneous full channel correlation matrix, we extend the correlation matrix distance (CMD) metric, originally defined for the narrowband fast fading scenario, to the wideband fast fading channel. The simulated CMDs from the NE-3GPP SCM are compared with that of the measurements taken in FLOWS project, which shows that: (1) The Kronecker assumption cannot be

1-4244-0330-8/06/$20.002006 IEEE

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

applied in calculating the instantaneous full channel correlation matrix, since it results in significant deficiencies in the simulated CMDs from the NE-3GPP SCM. (2) Using our further extensions to the 3GPP SCM, the wideband time variation of the modelled channels is closer to the variations observed in real situations than using the IB3G SCM.

(3) ))sin(2exp( ,,, nll tnAoDT

TnT j θxΨ π=

Here, the matrices and θ include the AoAs

and angles-of-departure (AoDs) within the nl tnAoA ,,θ

lnR,Ψ

nl tnAoD ,,

Tx

lnT ,

ln

Rx

th mid-path at time respectively. The receiver position is a 1-by- matrix, and the transmitter position is a 1-by- matrix. After calculation, and should be ( n )-by-K matrices.

nt Rn

Tn

TΨ RnII. EXTENSIONS TO THE 3GPP CHANNEL MODEL

A. The Generation of the Instantaneous Channel Matrix In the 3GPP SCM, there is a fixed number 6 of “paths” in every scenario, each of the paths being made up of 20 spatially separated “sub-paths”. In the IB3G SCM, since the property of intra-path delay-spread is added into the 3GPP SCM, the 20 sub-paths are split into subsets, denoted “mid-paths”, which are subject to different delays relative to the original path. To clarify the relations between the types of paths, we declare the definitions of several parameters before introducing the generation of the instantaneous channel matrix.

(4) ))cos(2exp( ,,,,, nlnll tnVtnAoAnRVnV tj θθxΨ −= π

where matrix includes the velocity values of receive antenna array, and thus should be a -by-1 matrix. The magnitude of the velocity is defined according to different scenarios [5]. In this simulation, the chosen velocity is 1.5 m/s. The matrix θ denotes the direction of the movement of the MS, which is a 1-by-K matrix. After these calculations, matrix should be a -by-K matrix [5].

RVx

Rn

Rn

nl tnV ,,

lnV ,Ψn , m, l one of the N paths, one of the M sub-paths, one of the L mid-paths in each path respectively

⋅⋅⋅= ntAoA,φθGθG

PΞ ,*)(*)(* lnl

l

jMSBS

nn e

Kdiag (5) ln the l th mid-path within the nth path

k one of the K sub-paths within each mid-path: note that K takes different values for different mid-paths

where φ is a matrix including instantaneous phases of

all the sub-paths within the nnl tnAoA ,,

ASGl

th mid-path arriving at the MS. and are antenna gains of the BS and MS, is

the power of the nBSG

lnP

lth mid-path. Therefore, Ξ should be a K-

by-K matrix with all of the power values on the diagonal positions.

ln

The urban macrocell scenario in [1] is considered here: therefore the number of mid-paths within each path is 3 and the total number of mid-paths is 18 according to the IB3G SCM [6]. The MIMO channel matrix for the wideband fast fading channels with transmit antennas and receive antennas is defined as:

Tn Rn

(1) TT,nnV,nR,nnnnn lllllTR

*t ΨΞΨΨΗ ⋅=)(,, To better generate the channel matrix for a wideband fast fading scenario, some extensions are made to the 3GPP SCM.

)(,, nnnn tlTR

H

where denotes the channel matrix for the n)(,, nnnn tlTR

H

nt

nt

( )T⋅

lth

mid-path at time (n = 1,2,3,...) [5], matrices and

denote the steering vectors for the lnR,Ψ

lnT ,Ψ

ln

ln

lnV ,

ln

th mid-path at

time at the receiver side and the transmitter side respectively, a uniform linear antenna array (ULA) is used for both transmitter and receiver side. Matrix stands for the

velocity vectors of the MS related to the

Ψth mid-path and

the diagonal matrix contains the sub-path amplitudes in

the lnΞ

th mid-path. In this paper, the following notations are defined as: symbol “ ⋅ ” means an element-by-element multiplication; stands for matrix transposition;

∗( )∗⋅ stands

for complex conjugation; stands for matrix Heimitian; denotes the expectation operator; denotes the trace

of a matrix; ⊗ denotes the Kronecker multiplication. The definitions of , , and Ξ are as follows:

( )H⋅

V ,Ψ

{}⋅E {}⋅

l

tr

nlnR,ΨlnT ,Ψ

ln

1) The different initial last bounce distances (LBDs) for the mid-paths

vθ

0,, tnAoA lθ

γ

1,, tnAoA lθ

0,, tnAoD lθ

subpath kcluster n

BS array

MS array

d

1tv ⋅

c

antenna array broadside

mid-path l

0,tnllbd

Figure 1: Angle parameters

(2) ))sin(2exp( ,,, nll tnAoAT

RnR j θxΨ π= Fig.1 shows the spatial structure of the angle parameters. The angles are all with respect to the broadside direction of the

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

antenna array, measured in an anti-clockwise direction. They have the same definitions as in the above section; however, some more parameters are defined as follows:

Step2: Determine . 1,, tnAoA l

θ

(12) )(1,, vtnAoA l

θπγθ −−=vθ direction of the movement of the mobile

This completes the calculation of the instantaneous AoAs for all the mid-path components, and obviously there will be K of the sub-paths at time correspond to the AoA of each mid-path.

1t 1,, tnAoA lθ

v MS velocity vector d, c distance between the last bounce scatterer of the ln th mid-path and the MS at time , t respectively 0t 1

γ angle between the direction of movement and direction of path arrival at time 1t 3) The time-variant phases of the multi-paths arriving at

the MS The initial LBDs of the paths are independent for all , and randomly generated from the lognormal

distribution, which was given in the IB3G SCM [6]. The distance between the last bounce scatterer of the

)...1( Nn =

ln th mid-path and the MS at time t is no more equal to the LBD of the n

0th path. They are generated on the basis of the LBD of the nth

path using the lognormal distribution, and can have small offsets relative to the nth path. The formulae are as follows:

The calculation of the instantaneous phases for all the multi-paths arriving at the MS, based on the calculation of the time variant AoAs, is as follows: Step1: We assume that the phases for the sub-paths within each mid-path at time t are drawn from a uniform random distribution on 0 to

0

π2 .

(6) Xdlbd tn += min, 0

(13) )2,0(~0,, πφ UniformtnAoA l

Step2: The phases of the sub-paths within each mid-path at time are determined by the following formula: 1t

)(10min md =

01 ,,,, tnAoAtnAoA ll

dc φλ

φ +−

= (14)

)1,22(~)log( 2

1

1 =−−

+= σττττ

µN

nGaussianX (7) where λ is the wave length.

where ~ means “is distributed as”, µ is the mean of the distribution and σ2 is its variance.

4) The time-variant mid-path powers Step1: The random average power for the nl

th mid-path at time can be generated from the parameters given by the IB3G SCM, so they are known as , each of them include the powers of K sub-path components [6].

0t

0,tnlP

(8) Ylbdlbd tntnl+=

00 ,,

)1,22(~)log( 2

1

1 =−

−+= σ

ττ

ττµ

nn

nn

L

lGaussianY (9) Step2: Determine the random average powers for the ln th mid-path at time based on the time variant time delays of the mid-paths at time .

1t

0twhere stands for the time delay of the

lnτ

0

ln th mid-path at

time . t

ctntn v

dcll

−+= 01 ,, ττ (15)

2) The instantaneous angles-of-arrival (AoAs) Since the initial values of the LBDs have been generated, the time variant AoAs can be generated using some trigonometric functions. Here we assume that: (1) the AoAs and AoDs at time are taken as a reference, so they are known; (2) the velocity and the direction of the MS are constants in these limited periods, and are also known. The algorithm for calculation of the AoAs for the

0t

ln th mid-path at time t is as follows:

1

where vc is the speed of light;

10/))(1(

', 10

1,11,

1lnDSDS

tntlnDS

lr

r

tn eP ξσ

ττ−

−−

= (16)

where ,DSr DSσ and are parameters generated from the formulae given by 3GPP SCM [1]. Note that the average powers are normalized so that the total average power for all the mid-paths is equal to one.

lnξ

Step1: Determine the parameter γ using the trigonometric functions.

∑×

=

= LN

ntn

tntn

ll

ll

P

PP

1

',

',

,

1

11

(17)

−−=

cdtv tnAoAv l

)cos(arccos 0,,1 θθ

γ (10)

The above calculation completes the extensions made to the 3GPP SCM to better generate the instantaneous wideband )cos(2)(

0,,122

1 tnAoAv ldtvdtvc θθ −−+= (11)

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

channel matrix. The channel matrix is now 4-dimensional, and can be written as

( ){ }TTRRnnTRnn ninitiitll

KK 1,1,,,,),( === ττ HH

lnτ

for given

time delay at each time instant tn. [3] In this paper, the channel impulse responses are transferred to frequency responses using Fourier transformation; hence the channel matrix for each frequency at each time can be generated, written as

III. EVALUATION OF THE SIMULATION BY MEASUREMENTS FROM THE FLOWS PROJECT

( ){ TTRRnTRn ninitfiitf KK 1,1,,,,),( === HH } . Therefore, it is more convenient to generate the instantaneous channel correlation matrix for each frequency, and this will be the prerequisite to generate the CMD for the wideband scenario [8].

The propagation measurements in the FLOWS project were performed during August and September 2002 in and around Oslo, Norway. The measurements were performed at a frequency of 2.1 GHz over a bandwidth of 50 MHz. The actual sites chosen included both indoor and outdoor scenarios, but considering that the 3GPP SCM is mainly designed for the outdoor scenario, the measurement used in this paper is performed in a downtown called “Kvadraturen” in Oslo [2].The spatial realization of the measurement is 88× MIMO system. The time spacing between two neighbouring snapshots is 200 ms. The narrowband instantaneous correlation matrix generated from the measurement is averaged over all the frequencies. B. The Instantaneous Full Channel Correlation Matrix of

the Wideband Channels IV. THE CMD METRIC IN WIDEBAND TIME VARIANT SCENARIO In this paper, the instantaneous full channel correlation

matrix, which is written as R , is calculated for each frequency at each time using two different methods; one uses the Kronecker assumption, the other does not. Under the Kronecker assumption, the algorithm for the calculation of

is defined as [7]:

),( ntf

),( ntfR

In [4] Herdin and Bonek have introduced the correlation matrix distance (CMD) as a metric for the variation of the spatial structure of spatially non-stationary narrowband fast fading MIMO channels. It is defined as:

2221

21

)()()}()({tr

1tttt

dcorr RRRR

−= (21) { } RxTx

Rxntf RR

RR ⊗=

tr1),( (18)

This definition is based on the inner product of two instantaneous correlation matrices and :

is this inner product, and )( 1tR )( 2tR

)}()({tr 21 tt RR 2⋅ denotes the Frobenius norm of the given matrix. When the correlation matrices are identical (apart from a scalar factor), the CMD is zero, while the maximum value is unity if the matrices are completely uncorrelated [5].

with the transmitter correlation matrix and receiver correlation matrix given by [7].

TxR

RxRTherefore, under the Kronecker assumption, the full channel correlation matrix can be seperated into the transmitter side and receiver side respectively, which has the big advantage that the entries of the full channel correlation matrix can be simplified from to independent elements. However, this simplification will result in less accurate modelling of the short-term time variation of the wideband channels, which will be analysed in the following sections.

( 2TRnn ) 22

TR nn + In this paper, the CMD metric is extended to apply to the wideband time variant MIMO channels. This wideband CMD metric is defined as:

f

n

f n

n

f

n

fncorr

ncorr n

tftftftftr

n

tfd

td

ff

∑∑==

−

== 1 221

1

1

]),(),()),(),((

1[),(

)(RRRR

(22) Without the Kronecker assumption, the algorithm for the calculation of is defined as [8]: ),( ntfR

( ) ( )( ){ }Hnnn tftftf ,vec),(vecE),( HHR = (19)

with given by ( ),(vec ntfH )

( ) ( )( )1,,,,reshape),(vec ,TRnTRn nntfnntf HH = (20)

where “reshape” is a function (equivalent to the MATLAB function of the same name) that returns a

vector whose elements are taken columnwise from matrix H .

1−− bynn TR

),( ntf

Here, in the wideband case, and are two instantaneous correlation matrices for a certain frequency f, and the CMD at each time instant is the averaged value over all the CMDs at all the frequencies. As for the narrowband case, if the instantaneous correlation matrices are identical for all the frequencies (apart from a scalar factor), the CMD is zero; while if they vary radically, it will tend to unity.

),( 1tfR ),( ntfR

Based on the above definition for the wideband case, the following questions can be investigated: one is that if the Kronecker assumption is able to characterise the short-term time variation of the wideband channels, and the other is that when modelling the wideband time variant channels if the performance of the NE-3GPP SCM is better than the IB3G SCM comparing with the FLOWS measurements. Some simulations have been done to answer these two questions, and the simulation results are shown in Fig.2 and Fig.3

The 17th Annual IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC'06)

respectively. For both of them, the simulation results are obtained from the MIMO system with 8 antennas at both sides arranged as a uniform linear array (ULA) with

λ5.0 spacing. In total, 384 time instants of the full channel correlation matrices are created; the time period between two neighbouring time instants is 20 ms. In Fig.2, the “*” and “o” lines show the CMDs generated with and without the Kronecker assumption respectively. It shows that, for the first 50 samples, the variation of the “*” line is very similar to the “o” line, but after sample 50, the variation of the “o” line is much closer to the measurement, and the amplitude of the “*” line is much lower than the measurement comparing with the “o” line, which implies that with the Kronecker assumption, the full channel correlation matrices lose much channel information when characterising the short-term time variation of the wideband channels. Therefore, we choose to calculate the full channel correlation matrix without the Kronecker assumption. In Fig.3, the CMDs from the NE-3GPP SCM are compared with that of the IB3G SCM. The “o” line and the “*” line are the CMDs generated from the NE-3GPP SCM and the IB3G SCM respectively. It shows that the CMD for the two lines both stay nearly zero for the first 5 samples and then they differ from each other. On the one hand, the CMD of the “*” line quickly increases to a value of 0.8, and after sample 40, it stays approximately constant, which means that the change of the correlation matrix has already reached its maximum. On the other hand, the correlation matrix distance of the “o” line increases to a value of 0.8 more slowly than the “*” line, from sample 80 to 150, it increases to 0.9 gradually, and then, after sample 150, it stays approximately constant. The correlation matrix distance for one of the measurements taken in FLOWS project is shown by the “.” line in Fig.3. In total 384 time samples are generated in this measurement. The measurements (“.” line) are closest to the simulation results for the NE-3GPP SCM (“o” line) than for the IB3G SCM (“*” line). The time variation of the measured wideband channel is not as fast as predicted by the IB3G SCM, but the amplitude of the variation is higher than that predicted by the IB3G SCM, which implies that with many more time variant parameters taken into account, the NE-3GPP SCM predicts time variations for the wideband channels which are close to actual measurements.

V. CONCLUSIONS

Following the extensions to 3GPP SCM in WINNER project, this paper proposes further extensions to the 3GPP SCM to investigate the time variation of the wideband channels in an outdoor scenario. By considering more parameters that influence the instantaneous channel matrix, and hence the channel correlation matrix, the NE-3GPP SCM can more realistically simulate the time variation of the wideband channels than the IB3G SCM. This has been demonstrated by comparison with the measurements taken in the FLOWS project by the extended CMD metric. It is also worth mentioning that the Kronecker assumption cannot be applied in calculating the full channel correlation matrix, or it will

lead to siginificant deficiencies in simulating the time variation of the wideband channels using the NE-3GPP SCM.

0 50 100 150 200 250 300 350 400-0.2

0

0.2

0.4

0.6

0.8

1

time sample

CM

D

dot : measurement star : with the Kronecker assumption circle : without the Kronecker assumption

Fiugre 2: MIMO wideband fast fading CMDs with and without the

Kronecker assumption

0 50 100 150 200 250 300 350 400-0.2

0

0.2

0.4

0.6

0.8

1

time sample

CM

D

dot : measurement star: IB3G SCM circle :NE-3GPP SCM

mean changing rate difference: 7.5%

mean amplitude difference: 10%

Figure 3: MIMO wideband fast fading CMDs from the IB3G SCM and the NE-3GPP SCM

REFERENCES [1] 3GPP TR 25.996 V6.1.0 (2003-09), Technical Report. [2] IST-2001-32125 Flows Deliverable Number: D10 Documentation of the

Measurement Campaign—Part 1:2.1 GHz. [3] Alister G. Burr, “Modulation and coding for wireless communications”,

Prentice-Hall, 2001. [4] Markus Herdin and Ernst Bonek “A MIMO Correlation Matrix based

Metric for Characterizing Non-Stationarity”, Institut für Nachrichtentechnik und Hochfrequenztechnik, Technische Universität Wien, Austria.

[5] H. Xiao and Alister G. Burr, "Simulation of time-selective environment by 3GPP spatial channel model and analysis on the performance evaluation by the CMD metric", Wireless Personal Multimedia Communications (WPMC) Conference, Aalborg, Denmark, September 2005.

[6] Daniel S. Baum and Jan Hansen, Giovanni Del Galdo and Marko Milojevic, Jari Salo, pekka Kyosti, “An Interim Channel Model for Beyond-3G Systems: Extending the 3GPP Spatial Channel Model (SCM)”, IEEE VTC 2005 Spring.

[7] Huseyin Ozcelik, Nicolai Czink, Ernst Bonek, “What Makes a Good MIMO Channel Model”, Insititut fur Nachrichtentechnik und Hochfrequenztechnik, Technische Universitat Wien, Vienna, Austria.

[8] Hui Xiao, Alister G. Burr and Lingyang Song, “A Time-Variant Wideband Spatial Channel Model Based on the 3GPP Model”, IEEE Vehicular Technology Conference, Montreal, Canada, September 2006.