A Multiple Input - Multiple Output Channel Model for Simulation of TX- and RX-Diversity Wireless Systems Matthias Stege, Jens Jelitto, Marcus Bronzel, Gerhard Fettweis Mannesmann Mobilfunk Chair for Mobile Communications Systems, Dresden University of Technology, Germany E-mail:
[email protected]
Abstract Space-time receivers for wireless communication systems offer the possibility to have both TX- and RX-antennas. For a realistic simulation of such systems, a multiple input multiple output (MIMO) spatial channel model is required which reasonably characterizes the space- and time-variant effects of the mobile radio channel. This paper describes a space-time vector channel model with realistic fading simulation for different scenarios. Mutual correlation between the fading coefficients is considered. This allows an estimation of the diversity gain, that can be achieved with spacetime receivers in different scenarios.
tors in the far-field. For each dominant reflector one significant multipath is assumed. This path consists of a large number of incoming waves which result from the structure of local scatterers in vinicity of the transmitter and receiver. Since the relative delays of these waves are small, they cannot be resolved by the receiver. In case of any movement in the scenario the superposition of the waves results in a space-time fading process. Independent fading is assumed for each significant multipath p with a specific time delay p . After the introduction a MIMO-signal model in Section 2 including a discrete time matrix formulation, the MIMOchannel characteristics and their modeling are described in Section 3 in more detail. In Section 4 the results of the space-time fading models are compared with theoretical assumptions.
1 Introduction 2 Signal Model In order to analyze the performance of new space-time concepts such as adaptive antennas, space-time processing and -coding techniques, an adequate space-time channel model is essential. Current vector channel models represent the spatial fading characteristics only at the receiver (RX) [4]. Although this channel models are sufficient for simulation of space-time concepts at the receiver, they are not directly applicable for scenarios, where space-time concepts are applied at the transmitter (TX). For the performance analysis of TX-diversity concepts, the correlation between the wireless channel from each TX- to each RX-antenna is essential. A realistic model of the correlation between fading coefficients at different TX-antennas is therfore necessary to evaluate the performance of TX-diversity concepts such as space-time coding [12] for different propagation scenarios. The presented MIMO-channel model allows a common simulation of TX as well as RX-space-time concepts. The MIMO-channel model is based on the assumption, that there are are few dominant spatially well separated reflec-
A space-time channel with M Tx transmit and M Rx receive antennas is considered. In general, the signals of the TXantennas are transmitted over M Tx different single input multiple output-channels (SIMO-channel). The vector
hm(t; ) = hm;0(t; ) hm;1(t; ) : : : hm;M
Rx
T
1 (t; )
(1)
represents the communication channel from transmit antenna m to every receive antenna. The received signal vector follows from a superposition of signals transmitted over M Tx antennas:
x(t) = i( )
Tx MX
m
1 X 1
=0 k= 1
sm (k ) hm (t; t
|
+i(t) + n(t);
where t represents the interference and vector.
kTsym ) {z
}
(2)
n(t) the noise
1 After sampling with a sampling period of Ts Tsym the oversampled stream of transmitted symbols from the m-th TX-antenna s"m t is defined as follows (The oversampling factor is assumed to be a non-negative integer.):
=
()
The complex path coefficients cp;m;r n are defined as:
cp;m;r (n) = Pp p (n) p;m;r (n)
()
s" (nTs )
=
Tsym
1 X k
= 1
s(k ) Æ (nTs
1)
()
kTsym ) (3) c0,m,r (t)
()
x(nTs ) =
Tx MX
m
1 X 1
=0 l= 1
c 1,m,r (t) c 2,m,r(t)
(4)
The sampling period Ts is dropped in all following discrete time representations for notational convinience.
-8
)
1
hm(n; n l) =
hm;0 (n; n
hm;r (n; n
l) =
1
=0
p
cp;m;r (n) g (n
l
= p + Rp;r + Tp;m: x
x
6
8
10
12
14
16
τ /Ts
ηN0
( )
P L( p )[dB] = Pref
+ 10n log( p = ref ): (9) Path loss exponents 2 < n < 6 are possible depending on the propagation scenario to be considered.
Alternatively, characteristic power-delay profiles may be used, which are defined for ITU channel models.
T
l)
Due to movement there are fading effects, that affect the instantaneous received power of a path p. They can be classifies as slow and fast fading. Slow fading (shadowing) is often neglected for bit level simulations of digital communication system since a long simulation time would be consumed to achieve statistic at reliable results for this fading process. Therefore, slow fading is assumed to be constant over the simulation time for the bit level channel model.
(5)
p;m;r ) (6)
Paths that arrive with small delays to the significant multipaths result from the local scattering structure in the vicinity of the transmitter, dominant reflectors or receiver. In case of any movement this local scattering results in a space-timevariant fading processes. The space-time fading model is described in Section 3.
The delay p;m;r consist of the path delay p and the propagation delays across the TX- and RX-antenna arrays:
p;m;r
4
Nτ
=
()
P X
2
x It is assumed that p Rp;r ; Tp;rx and therefore the averp P L p is constant for all TX- and age path loss Pp RX-antennas. The mean power of each multipath component depends on the overall propagation delay p and is defined as [10]
The channel vector is defined as the sum of P convolutions of the channel coefficients cp;m;r n for the path p, TXantenna m and RX-antenna r with the pulse shaping filter response g of the communication system.
()
0
h( )
12
l) : : : hm;M Rx 1 (n; n
-2
Figure 1. Channel impulse response t; for the multipath channel from TX-antenna m and RXantenna r
( )
)
-4
N
Figure 1 shows the channel impulse response hm;r t; of the communication channel between TX-antenna m and RX-antenna r. The discrete time representation of the channel vector n; n l follows from (1):
-6
η N0
Following the characteristics of the channel impulse response m n; n l for a multipath environment are described in more detail. The transmitted signal arrives at the receiver with some delay 0 . Additional to the first path, there are echos coming from reflections, diffraction or scattering of the signal at these objects. Significant multipaths appear from dominant remote reflectors with a significant distance to the receiver. The number of such significant multipaths (P ) depends strongly on the propagation scenario. Usually the number of significant multipath are as[2]. The ability of sumed to be in the range of P a receiver to resolve multipath components depends on the bandwidth of the communication system.
h(
hm,r(t, τ )
s"m (lTs ) hm (nTs ; (n l)Ts )
+i(nTs) + n(nTs)
h (
()
with the average pathloss Pp , p n describes the slow fading (shadowing) and p;m;r n the space-time fast fading.
Oversampling of the transmitted signal is done by inserting zeros between each symbol sm k . The discrete time representation of (2) follows:
(
(8)
(7) 2
In matrix formulation (4) can be expressed as:
Remote Scatterer p
x(n) = H(n) s"(n) + i(n) + n(n): )
]
1
H ()
H(n) = H1(n) H2(n) : : : HM (n) ; (11) where Hm (n) defines the channel through which the signal
Tx
Local Scattering
p
φTx p
Tx
M
th
The N M Tx M Rx - dimensional channel matrix contains of M Tx SIMO-channels m n ,
pa
[(
(10)
φTX 0
is transmitted from TX-antenna m to all RX-antennas:
LOS
−∆
1
φp
Rx
φRX 0
TX-Antennas (i.e. BS)
Hm(n) = hm(n; 0) hm(n; 1) : : : hm(n; N 1):
∆
(12)
M
RX-Antennas (i.e. MT)
The signal vector consists of N subsequent samples of the signals from each of the M Tx TX-antennas: h
s"(n) = s"0 (n) : : : s"0 (n
N + 1) : : :
s"M Tx 1 (n) : : : s"M Tx 1 (n
i( )
Furthermore, n and in a similar manner.
Figure 2. Geometry of the scattering scenario. iT
N + 1) :
(13)
Since the characteristic AOA and AOD of a path depend on the position of the dominant reflector they are somehow associated with the delay of that path. It is reasonable to expect that significant multipath components which arrive within a short delay interval tend to be clustered and have therefore a similar AOA’s and AOD’s. One approach which takes this into account is to define a certain dominant spatial reflector distribution and to derive the spatial channel characteristics from this distribution, e.g. [5]. For simplicity, these models use the single bounce assumption, which means, that each multipath component is created by a specular reflection of the wavefronts at the reflecting object.
n(n) contain interference and noise
The fading characteristic of the MIMO-channel and the mutual correlation of the space-time-fading processes is important for the receiver performance in a specific scenario. In Section 3 the spatial characteristics are described in more detail, since they determine the characteristics of the spacetime-fading.
3 MIMO Channel Model
Only uniform linear arrays (ULA) are considered in this paper. The array propagation vector defines the spatial response of an antenna array. It is assumed that the RXantennas are within the far-field of the TX-antennas. Therefore, the plane wave assumption holds for all RX-antennas. The propagation of a plane wave representing the path p x impinging on the antenna array causes a time delay Rp;r at different antenna elements. This small time delay of the arrival of the wavefronts between different antennas results in a phase-shift r;p at these RX-antennas:
For each significant multipath component p the angle of arrival (AOA) Rp x and the angle of departure (AOD) Tp x are defined with respect to the array normal of the according antenna array and the position of a dominant reflector. The orientation of the TX-array and the RX-array with respect to the line of sight (LOS) is described by T0 X and RX 0 , respectively (Figure 2). This model does not consider elevation angles.
( )
Due to local scattering there is some angular spread p Rp x for each significant multipath p, which results in replica of the signal having almost the same delay p , but arise from different AOA’s Rp;lx : v u u t
1 LX1( R )2 p ( p ) = p;l L Rx
x
=0
l
1 LX1 R
L l=0
x
p;l
Rx
Rrp = r ( Rp ) = 2Rp;r c : x
x
x
(15)
x at RX-antenna r compared to the first anThe delay Rp;r tenna is defined as:
R R Rp;r = r d csin p
!2
x
x
(14)
x
;
(16)
where dRx is the distance between two adjacent antennas and is the carrier wavelength of the communications system.
( ) can be defined for local
A similar angular spread p p scattering at the transmitter.
Tx
3
a
The array propagation vector Rp x contains these phase shifts with respect to the first antenna for a certain path p. For a uniform linear array with antenna spacing dRx this vector Rx p can be expressed as:
a
ap = 1 Rx
e j 1;p Rx
:::
e
x T j R M Rx 1;p :
follows from the relative large distance between the dominant reflector and the receiver. It causes a temporal- , but no spatial-fading at the receiver. However, local scattering in vicinity of the receiver (RX2 in Figure 3) results in larger angular spreads, since the distance between the RXantennas and the local scatterers is small. Therefore, local scattering in the vicinity of the receiver results in a spacetime fading process.
(17)
and describes the spatial response of the array to a waveform impinging from direction Rp x . The same principle also applies to the TX-antenna array where the array response vector defines the relative phase shifts of the signals arriving from different antennas of the TX-antenna array at a dominant reflector:
ap = 1
Tx
e
x j T 1;p
T j Tx e M Tx 1;p ;
:::
TX
(18) RX2
with:
Tm;p = 2Tp;m c : x
x
(19)
RX1
for the m-th TX-antenna and the p-th path. Figure 3. Different local scattering scenarios at the receiver
Due to the limited time resolution not all incoming echos of the signal can be separated. In case of any movement, the path length to each local scatterer changes, resulting in a time varying complex fading process. For a given velocity v , the maximum frequency shift is fd fc v=c for a carrier frequency fc . The fading process p;m;r t of the p-th path between the m-th TX-antenna and the r-th RX-antenna is a superposition of L incoming local scattering wavefronts:
=
p;m;r (t) =
p1
1
L X
L l=0
With a variation of the distribution of Rp;lx different spatial scenarios can be simulated. Further the effects of the antenna displacement on the correlation can be studied with this model. It is therefore suitable for simulating different space-time receiver concepts such as diversity as well as beamforming approaches.
()
am ( Tp;lx ) ar ( Rp;lx ) p;l e j 2fd t cos( p;l ) Tx
= p;m;r (t) e j
p;m;r
4 Simulation Results
(20)
(t)
(21)
4.1 SIMO-Space-Time Fading
Each signal arriving from these local scatterers undergoes an attenuation p;l which is assumed to result from a ran. The AOD is often modeled as dom process with a uniform random distribution Tp;lx which yields the well-known Jakes-Spectrum [3].
=1
()
0
To study the effects of space-time-fading, first a single TXantenna (SIMO-channel) is considered. The correlation of the fading coefficients at the RX-array is a measure for the spatial fading characteristic of the SIMO-channel and depends on the displacement dRx of the antennas and the angular distribution. The spatial fading correlation of path p :
2
()
x t give rise to the typical While p;m;r t and Tp;m;r temporal correlation characteristic of the fast fading process, the two array propagation phase shifts am Tp;lx and ar Rp;lx define the spatial characteristics of the fading process.
( )
( )
Rs;p = Rx
Significant multipaths resulting from largely spaced dominat reflectors have different resolvable delays p . The fading process for each significant multipath is assumed to be independent.
1
L X
=0
l
a( p;l )aH ( p;l ) Rx
Rx
(22)
can be estimated for a uniform distribution of Rp;lx within Rp;lx Rp x Rx as given in [6]. Figure 4 and 5 shows the correlation of the fading signals at two antenna elements as a function of their separation dRx = for different AOA’s ( Rp x ) and angular spreads. Obviously small angular spreads
=
Wavefronts that result from local scattering only around the transmitter arrive from a distinct angle of arrival Rp x having only a small angular spread (RX1 in Figure 3). This 4
1.2
1.2
=0
o
p
Spatial Envelope Correlation s
Spatial Envelope Correlation s
Rx
1
=0 =5 = 15 = 60 = 180 o
o
0.8
o
o
0.6
o
0.4
0.2
0 0
0.5
1
1.5
2
2.5
3
3.5
4
Spatial Sensor Separation d=
4.5
5
Figure 4. Spatial correlation for different angular spreads and a AOA= o
x
= 60
o
1
=0 =5 = 15 = 60 = 180 o
o
0.8
o
o
o
0.6
0.4
0.2
0 0
0.5
1
1.5
2
2.5
3
3.5
Spatial Sensor Separation d=
4
4.5
Figure 5. Spatial correlation for different angular spreads and a AOA= o
0
R
Rx
60
()
cesses p;m;r t for each of the M Tx TX-antennas.
results in a high spatial correlation.
High correlation of the fading coefficients at different RXantennas can be observed in uplink urban scenarios with small angular spreads, which is the case if the base station is located at the roof tops and the mobile moves on the street level. In this case all incoming L local reflections arrive Rp x ; 8 l). The from approximately the same AOA ( Rp;lx fading coefficients at different RX-antennas differ only by a phase shift that is defined by the array propagation vector Rp x (17) determined by the distinct AOA of the multipath.
Rs;p = Tx
=
1
L X
=0
l
a( p;l )aH ( p;l ) Tx
Tx
(23)
Similar to the spatial correlation at the RX-array, a high correlation can be observed for low local angular spreads at the transmitter and vice versus.
a( )
( )
( )
The angular spreads p Rp x and p Tp x due to local scattering in vicinity of the TX- and RX-antennas are assumed to be independent. This is the case, if the antenna heights are not the same which results in a different spatial scenario at the transmitter and receiver (eg. the angular spread at a base station antenna on roof top level can be assumed to be much smaller than for an antenna operating on street level with local scattering from many surrounding objects)
Significant multipaths resulting from largely spaced dominat reflectors have different resolvable delays p . The fading process for each significant multipath is assumed to be independent. For downlink urban scenarios, the local angular spread at the RX-antennas is higher than for the uplink, because of the small distance of local scatterers to the antennas. Therefore, fading at the antennas becomes less correlated due to the increased angular spread. This holds even for small antenna displacements dRx .
Hence, a scenario with considerable space diversity at the RX-as well as TX-antennas would be a base station below rooftop and a mobile operating in a bad urban environment. In this case the RX- and the TX-spatial correlations are both low and diversity can be gained using both TX- and RXdiversity concepts. Spatially independent fading processes can be observed at the antenna array even when the antennas are spaced only a few wavelengths or less.
4.2 MIMO-Space-Time Fading For a MIMO-channel, the spatial fading correlation between signals from different TX-antennas has to be modeled as well. This correlation depend on the distribution of the AOD’s Tp;lx and results in mutual correlated fading pro-
Measurements suggest that it might be reasonably to assume a Laplacian rather than a uniform distribution for local AOA’s in uplink scenarios [7, 8]. The temporal fading characteristic depends on the number of local scatterers L 5
5
1
1
0.9
0.9 Theory RX 0.8
0.8
Correlation at RX Antennas
Theory TX Correlation at TX Antennas
Tx
RX-antenna Spacing d = 0.5 λ
Spatial Correlation ρ
Spatial Correlation ρ
0.7
0.7
0.6
0.5
0.4
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0.5
1
1.5
2
2.5
3
antenna displacement d/ λ
3.5
0
4
Tx
0.6
0.3
0
Antenna Spacing d = 5λ
0
2
4
6
8
10
12
14
16
18
20
antenna displacement d/λ
Figure 6. MIMO-Fading Correlation between a signal received at different RX-antennas
Figure 8. MIMO-Fading Correlation between signal transmitted from different TX-antennas
Remote reflector
the theoretical results. The small differences arrise from the finite number of local scatterers L and the limited observation time. This shows that the modeling of the MIMOfading process (20) have approximatly the same correlation characteristic as the theory.
∆Rx
Rx
∆
−∆Rx
5 Conclusion RX local scattering
TX local scattering
In this paper a multiple-input multiple-output (MIMO) channel model has been derived. Such a model is essential for realistic bit-level simulations of communication systems that use space-time-codes and other concepts that exploit spatial diversity. The main advantage of the presented model is the realistic space-time fading simulation at the receiver as well as the transmitter. In the past, the performance of space-time codes and other TX-diversity techniques were simulated for uncorrelated fading. With this channel model a more realistic simulation can be made, which it allows investigations of the actually achievable diversity gains using antenna arrays for a wide range of scenarios.
Figure 7. MIMO fading scenario
as well as on the distribution of Tp;lx . The widely used assumption of a uniform distribution which results in the wellknown Jakes-Spectrum [1, 3] is achieved in the limit of an infinite number of local scatters and a uniform distribution of Tp;lx : This theoretical assumption is reasonable for simulations, since it defines a worst case scenario for the space-time fading that might occur in a real environment. Figure 6 and 8 shows the correlation of simulated spacetime fading processes for 8 RX-antennas and 4 TX-antennas compared to theoretical results from [6]. In this example local scatterers with a uniform distribution of L
Tp;x0
Future work will consider the statistical description of spatial scatterer distributions for the significant multipaths components and the local scattering. Furthermore, the difference of the spatial scatterer distributions for different antenna heights needs to be examined in more detail. Measurements should be made to comply the results.
= 100 = 20o; T = 50o; Rp;0 = 50o; T = 180o x
x
x
was assumed. The simulated correlation matches almost 6
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