21-05-2012, 11:27 AM
MIMO Wireless Channels: Capacity and Performance Prediction
[attachment=22527]
Abstract
We present a new model for multiple input multiple-output (MIMO) out door
wireless fading channels which is more general and realistic than the usual i.i.d. model.
We investigate the channel capacity as a function of parameters such as the local
scattering radius at the transmitter and the receiver, the distance between the transmit
(TX) and receive (RX) arrays, and the antenna beamwidth sand spacing. We point out
the existence of “pinhole" channels which exhibit low fading correlation between
antennas but still have poor rank properties and hence low capacity. Finally we show
that even at long ranges high channel rank can easily be obtained under mild scattering
conditions.
1. INTRODUCTION
Multiple-input multiple-output (MIMO) communication techniques make use of multielement
antenna arrays at both the TX and the RX side of a radio link and have been shown
theoretically to drastically improve the capacity over more traditional single-input multiple output
(SIMO) systems [2, 3, 5, 7]. SIMO channels in wireless networks can provide diversity gain, array
gain, and interference canceling gain among other benets. In addition to these same advantages,
MIMO links can offer a multiplexing gain by opening Nmin parallel spatial channels, where Nmin is
the minimum of the number of TX and RX antennas. Under certain propagation conditions
capacity gains proportional to Nmin can be achieved [8]. Space-time coding [14] and spatial
101seminartopics.com
multiplexing [1, 2, 7, 16] (a.k.a. \BLAST") are popular signal processing techniques making use of
MIMO channels to improve the performance of wireless networks. Previous work and open
problems. The literature on realistic MIMO channel models is still scarce. For the line-of-sight
(LOS) case, previous work includes [13]. In the fading case, previous studies have mostly been
conned to i.i.d. Gaussian matrices, an idealistic assumptions in which the entries of channel
matrix are independent complex Gaussian random variables [2, 6, 8]. The influence of spatial
fading correlation on either the TX or the RX side of a wireless MIMO radio link has been
addressed in [3, 15]. In practice, however, the realization of high MIMO capacity is sensitive not
only to the fading correlation between individual antennas but also to the rank behavior of the
channel. In the existing literature, high rank behavior has been loosely linked to the existence of a
dense scattering environment. Recent successful demonstrations of MIMO technologies in
indoor-to-indoor channels, where rich scattering is almost always guaranteed. Confirms this [9].
Despite this progress, several important questions regarding outdoor MIMO channels remain
open and are addressed in this paper:_
What is the capacity of a typical outdoor MIMO channel? _
What are the key propagation parameters governing capacity? _
Under what conditions do we get a full rank MIMO channel (and hence high capacity)? _
What is a simple analytical model describing the capacity behavior of outdoor MIMO wireless
channels ?
Here we suggest a simple classification of MIMO channel and devise a MIMO channel model
whose generality encompasses some important practical cases. Unlike the channel model used
in [3, 15], our model suggests that the impact of spatial fading correlation and channel rank are
decoupled although not fully independent, which allows for example to describe MIMO channels
with uncorrelated spatial fading at the transmitter and the receiver
but reduced channel rank (and hence low capacity). This situation typically occurs when the
distance between transmitter and receiver is large. Furthermore,our model allows description of
MIMO channels with scattering at both the transmitter and the receiver. We use the new model to
describe the capacity behavior as a function of the wavelength, the scattering radii at the
transmitter and the receiver, the distance between TX and RX arrays, antenna beamwidths, and
antenna spacing. Our model suggests that full MIMO capacity gain can be achieved for very
realistic values of scattering radii, antenna spacing and range. It shows, in contrast to usual
intuition, that large antenna spacing has only limited impact on capacity under fairly general
conditions. Another case described by the model is the “pin-hole" channel where spatial fading is
uncorrelated and yet the channel has low rank and hence low capacity.We show that this
situation typically occurs for very large distances between transmitter and receiver. In the 1 * 1
101seminartopics.com
case (i.e. one TX and one RX antenna), the pinhole channel yields capacities worse than the
traditional Rayleigh fading channel. Our results are validated by comparing with a ray tracingbased
channel simulation. We find a good match between the two models over a wide range of
situations.
2. CAPACITY OF MIMO CHANNELS AND MODEL
CLASSIFICATION
We briefly review the capacity formula for MIMO channels and present a classification of MIMO
channels. We restrict our discussion to the frequency-flat fading case and we assume that the
transmitter has no channel knowledge whereas the receiver has perfect channel knowledge.
2.1. Capacity of MIMO channels
We assume M RX and N TX antennas. The capacity in bits/sec/Hz of a MIMO channel under an
average transmitter power constraint is given by1 [2]
C = log2 [det (_IM +ρ/N HH +)] (1)
Where H is the M * N channel matrix, IM denotes the identity matrix of size M; and _ is the
average signal to-noise ratio (SNR) at each receiver branch. The elements of H are complex
Gaussian with zero mean and unit variance, i.e., [H]m;n _ CN(0; 1) for m =1; 2; :::;M; n = 1;2; :::;N.
Note that since H is random C will be random as well. Assuming a piece-wise constant fading
model and coding over many independent fading intervals2, EHfCg can be interpreted as the
Shannon capacity of the random MIMO channel [5].
1The superscript _ stands for Hermitian transpose.
2EH stands for the expectation over all channel realizations.
2.2. Model classification
Let us next introduce the following MIMO theoretical channel models:
Uncorrelated high rank (UHR, a.k.a. i.i.d.) model:The elements of H are i.i.d. CN(0; 1). This is the
dealistic model considered in most studies.
Uncorrelated low rank (ULR) (or \pin-hole") model: H = grx g_tx, where grx and gtx are independent
RX and TX fading vectors with i.i.d. complex-valued components grx
101seminartopics.com
CN(0; IM); gtx _ CN(0;IN). Every realization of H has rank 1 and therefore although diversity is
present capacity will be much less than in the ULR model since there is no multiplexing gain.
Correlated low rank (CLR) model: H =grxg_txurxu_tx where grx _ CN(0; 1) and gtx
CN(0; 1) are independent random variables and urx and utx are _xed deterministic vectors of
sizeM * 1
and N * 1, respectively, and with unit modulus entries.
This model yields RX array gain only.
1* 1 HR, de_ned by the UHR model with M = N = 1, also known as Rayleigh fading channel.
1* 1 LR, de_ned by the ULR or CLR model with M = N = 1 (double Rayleigh channel).
Note that the low rank models (ULR, CLR, 1_1 LR)above do not use the traditional normal
distribution for the entries of H but instead the product of two Gaussian variables. This type of
distribution is shown later to occur in important practical situations. In the 1*1 case,The LR model
has worsened fading statistics. This is due to the intuitive fact that a double Rayleigh channel will
fade \twice as often" as a standard Rayleigh channel [4].
3. DISTRIBUTED SCATTERING MIMO MODEL
We consider non-line-of-sight channels, where fading is induced by the presence of
scatterers at both ends of the radio link. The purpose is to develop a general stochastic channel
model that captures separately the diversity and rank properties and that can be used to predict
practically the high rank region of the MIMO channel.The particular case of LOS channels is
addressed in [4], where the authors derive a simple rule redicting the high rank region. In the
following, for the sake of simplicity, we consider the effect of near field scatterers only. We ignore
remote scatterers assuming that the path loss will tend to limit their contribution to the total
channel energy. Finally, we consider a frequency-flat fading channel.
3.1. SIMO Fading Correlation Model
101seminartopics.com
We consider a linear array of M omni-directional RX antennas with spacing dr. A number
of distributed scatterers act as perfect omnidirectional scatterers of a signal, which eventually
impinges on the RX array. The plane-wave directions of arrival (DOAs) of these signals span an
angular spread of θr radians (see Fig 1).
Figure 1: Propagation scenario for SIMO fading correlation. Each scatterer transmits a planewave
signal to a linear array several distributions can be considered for the DOAs, including
uniform, Gaussian, Laplacian etc. [10, 11]. The addition of different plane-waves causes space
selective fading at the RX antennas. It is well known that the resulting fading correlation is
governed by the angle spread, the antenna spacing and the wavelength. The RX array response
vector h can now be modeled as
(2)
where R_θr,dr is the M * M correlation matrix. For uniformly distributed DOAs, we find [10, 12]
where S (odd) is the number of scatterers with corresponding DOAs θr. For “large" values of the
angle spread and/or antenna spacing, Rθr,dr will converge to the identity matrix, which gives
uncorrelated fading. For “small" values of θr,dr, the correlation matrix becomes rank deficient
101seminartopics.com
(eventually rank one) causing(fully) correlated fading. For the sake of simplicity,we furthermore
assume the mean DOA to be orthogonal to the array (bore-sight). Note that the model provided in
(2) can readily be applied to an array of TX antennas with corresponding antenna spacing and
signal departure angle spread.
3.2. MIMO Correlated Fading Model
We consider the NLOS propagation scenario depicted in
Fig. 2.
Figure 2: Propagation scenario for fading MIMO channel.
The propagation path between the two arrays is obstructed on both sides of the link by a set of
significant near field scatterers (such as buildings and large objects) referred to as TX or RX
scatterers. Scatterers are modeled as omni-directional ideal reflectors. The extent of the
scatterers from the horizontal axis is denoted as Dt and Dr, respectively. When omni-directional
antennas are used Dt and Dr correspond to the TX and RX scattering radius, respectively. On the
RX side, the signal reflected by the scatterers onto the antennas impinge on the array with an
angular spread denoted by θr, where θr is function of the position of the array with respect to the
scatterers. Similarly on the TX side we defined an angular spread θt. The scatterers are
assumed to be located su_ciently far from the antennas for the planewave assumption to hold.
We furthermore assume that Dt;Dr _R (local scattering condition).
[attachment=22527]
Abstract
We present a new model for multiple input multiple-output (MIMO) out door
wireless fading channels which is more general and realistic than the usual i.i.d. model.
We investigate the channel capacity as a function of parameters such as the local
scattering radius at the transmitter and the receiver, the distance between the transmit
(TX) and receive (RX) arrays, and the antenna beamwidth sand spacing. We point out
the existence of “pinhole" channels which exhibit low fading correlation between
antennas but still have poor rank properties and hence low capacity. Finally we show
that even at long ranges high channel rank can easily be obtained under mild scattering
conditions.
1. INTRODUCTION
Multiple-input multiple-output (MIMO) communication techniques make use of multielement
antenna arrays at both the TX and the RX side of a radio link and have been shown
theoretically to drastically improve the capacity over more traditional single-input multiple output
(SIMO) systems [2, 3, 5, 7]. SIMO channels in wireless networks can provide diversity gain, array
gain, and interference canceling gain among other benets. In addition to these same advantages,
MIMO links can offer a multiplexing gain by opening Nmin parallel spatial channels, where Nmin is
the minimum of the number of TX and RX antennas. Under certain propagation conditions
capacity gains proportional to Nmin can be achieved [8]. Space-time coding [14] and spatial
101seminartopics.com
multiplexing [1, 2, 7, 16] (a.k.a. \BLAST") are popular signal processing techniques making use of
MIMO channels to improve the performance of wireless networks. Previous work and open
problems. The literature on realistic MIMO channel models is still scarce. For the line-of-sight
(LOS) case, previous work includes [13]. In the fading case, previous studies have mostly been
conned to i.i.d. Gaussian matrices, an idealistic assumptions in which the entries of channel
matrix are independent complex Gaussian random variables [2, 6, 8]. The influence of spatial
fading correlation on either the TX or the RX side of a wireless MIMO radio link has been
addressed in [3, 15]. In practice, however, the realization of high MIMO capacity is sensitive not
only to the fading correlation between individual antennas but also to the rank behavior of the
channel. In the existing literature, high rank behavior has been loosely linked to the existence of a
dense scattering environment. Recent successful demonstrations of MIMO technologies in
indoor-to-indoor channels, where rich scattering is almost always guaranteed. Confirms this [9].
Despite this progress, several important questions regarding outdoor MIMO channels remain
open and are addressed in this paper:_
What is the capacity of a typical outdoor MIMO channel? _
What are the key propagation parameters governing capacity? _
Under what conditions do we get a full rank MIMO channel (and hence high capacity)? _
What is a simple analytical model describing the capacity behavior of outdoor MIMO wireless
channels ?
Here we suggest a simple classification of MIMO channel and devise a MIMO channel model
whose generality encompasses some important practical cases. Unlike the channel model used
in [3, 15], our model suggests that the impact of spatial fading correlation and channel rank are
decoupled although not fully independent, which allows for example to describe MIMO channels
with uncorrelated spatial fading at the transmitter and the receiver
but reduced channel rank (and hence low capacity). This situation typically occurs when the
distance between transmitter and receiver is large. Furthermore,our model allows description of
MIMO channels with scattering at both the transmitter and the receiver. We use the new model to
describe the capacity behavior as a function of the wavelength, the scattering radii at the
transmitter and the receiver, the distance between TX and RX arrays, antenna beamwidths, and
antenna spacing. Our model suggests that full MIMO capacity gain can be achieved for very
realistic values of scattering radii, antenna spacing and range. It shows, in contrast to usual
intuition, that large antenna spacing has only limited impact on capacity under fairly general
conditions. Another case described by the model is the “pin-hole" channel where spatial fading is
uncorrelated and yet the channel has low rank and hence low capacity.We show that this
situation typically occurs for very large distances between transmitter and receiver. In the 1 * 1
101seminartopics.com
case (i.e. one TX and one RX antenna), the pinhole channel yields capacities worse than the
traditional Rayleigh fading channel. Our results are validated by comparing with a ray tracingbased
channel simulation. We find a good match between the two models over a wide range of
situations.
2. CAPACITY OF MIMO CHANNELS AND MODEL
CLASSIFICATION
We briefly review the capacity formula for MIMO channels and present a classification of MIMO
channels. We restrict our discussion to the frequency-flat fading case and we assume that the
transmitter has no channel knowledge whereas the receiver has perfect channel knowledge.
2.1. Capacity of MIMO channels
We assume M RX and N TX antennas. The capacity in bits/sec/Hz of a MIMO channel under an
average transmitter power constraint is given by1 [2]
C = log2 [det (_IM +ρ/N HH +)] (1)
Where H is the M * N channel matrix, IM denotes the identity matrix of size M; and _ is the
average signal to-noise ratio (SNR) at each receiver branch. The elements of H are complex
Gaussian with zero mean and unit variance, i.e., [H]m;n _ CN(0; 1) for m =1; 2; :::;M; n = 1;2; :::;N.
Note that since H is random C will be random as well. Assuming a piece-wise constant fading
model and coding over many independent fading intervals2, EHfCg can be interpreted as the
Shannon capacity of the random MIMO channel [5].
1The superscript _ stands for Hermitian transpose.
2EH stands for the expectation over all channel realizations.
2.2. Model classification
Let us next introduce the following MIMO theoretical channel models:
Uncorrelated high rank (UHR, a.k.a. i.i.d.) model:The elements of H are i.i.d. CN(0; 1). This is the
dealistic model considered in most studies.
Uncorrelated low rank (ULR) (or \pin-hole") model: H = grx g_tx, where grx and gtx are independent
RX and TX fading vectors with i.i.d. complex-valued components grx
101seminartopics.com
CN(0; IM); gtx _ CN(0;IN). Every realization of H has rank 1 and therefore although diversity is
present capacity will be much less than in the ULR model since there is no multiplexing gain.
Correlated low rank (CLR) model: H =grxg_txurxu_tx where grx _ CN(0; 1) and gtx
CN(0; 1) are independent random variables and urx and utx are _xed deterministic vectors of
sizeM * 1
and N * 1, respectively, and with unit modulus entries.
This model yields RX array gain only.
1* 1 HR, de_ned by the UHR model with M = N = 1, also known as Rayleigh fading channel.
1* 1 LR, de_ned by the ULR or CLR model with M = N = 1 (double Rayleigh channel).
Note that the low rank models (ULR, CLR, 1_1 LR)above do not use the traditional normal
distribution for the entries of H but instead the product of two Gaussian variables. This type of
distribution is shown later to occur in important practical situations. In the 1*1 case,The LR model
has worsened fading statistics. This is due to the intuitive fact that a double Rayleigh channel will
fade \twice as often" as a standard Rayleigh channel [4].
3. DISTRIBUTED SCATTERING MIMO MODEL
We consider non-line-of-sight channels, where fading is induced by the presence of
scatterers at both ends of the radio link. The purpose is to develop a general stochastic channel
model that captures separately the diversity and rank properties and that can be used to predict
practically the high rank region of the MIMO channel.The particular case of LOS channels is
addressed in [4], where the authors derive a simple rule redicting the high rank region. In the
following, for the sake of simplicity, we consider the effect of near field scatterers only. We ignore
remote scatterers assuming that the path loss will tend to limit their contribution to the total
channel energy. Finally, we consider a frequency-flat fading channel.
3.1. SIMO Fading Correlation Model
101seminartopics.com
We consider a linear array of M omni-directional RX antennas with spacing dr. A number
of distributed scatterers act as perfect omnidirectional scatterers of a signal, which eventually
impinges on the RX array. The plane-wave directions of arrival (DOAs) of these signals span an
angular spread of θr radians (see Fig 1).
Figure 1: Propagation scenario for SIMO fading correlation. Each scatterer transmits a planewave
signal to a linear array several distributions can be considered for the DOAs, including
uniform, Gaussian, Laplacian etc. [10, 11]. The addition of different plane-waves causes space
selective fading at the RX antennas. It is well known that the resulting fading correlation is
governed by the angle spread, the antenna spacing and the wavelength. The RX array response
vector h can now be modeled as
(2)
where R_θr,dr is the M * M correlation matrix. For uniformly distributed DOAs, we find [10, 12]
where S (odd) is the number of scatterers with corresponding DOAs θr. For “large" values of the
angle spread and/or antenna spacing, Rθr,dr will converge to the identity matrix, which gives
uncorrelated fading. For “small" values of θr,dr, the correlation matrix becomes rank deficient
101seminartopics.com
(eventually rank one) causing(fully) correlated fading. For the sake of simplicity,we furthermore
assume the mean DOA to be orthogonal to the array (bore-sight). Note that the model provided in
(2) can readily be applied to an array of TX antennas with corresponding antenna spacing and
signal departure angle spread.
3.2. MIMO Correlated Fading Model
We consider the NLOS propagation scenario depicted in
Fig. 2.
Figure 2: Propagation scenario for fading MIMO channel.
The propagation path between the two arrays is obstructed on both sides of the link by a set of
significant near field scatterers (such as buildings and large objects) referred to as TX or RX
scatterers. Scatterers are modeled as omni-directional ideal reflectors. The extent of the
scatterers from the horizontal axis is denoted as Dt and Dr, respectively. When omni-directional
antennas are used Dt and Dr correspond to the TX and RX scattering radius, respectively. On the
RX side, the signal reflected by the scatterers onto the antennas impinge on the array with an
angular spread denoted by θr, where θr is function of the position of the array with respect to the
scatterers. Similarly on the TX side we defined an angular spread θt. The scatterers are
assumed to be located su_ciently far from the antennas for the planewave assumption to hold.
We furthermore assume that Dt;Dr _R (local scattering condition).