1
Fundamental overview and simulation of MIMO systems for Space-Time coding and Spatial Multiplexing
Hoo-Jin Lee1, Shailesh Patil2, and Raghu G. Raj3
Wireless Networking and Communications Group (WNCG)
Dept. of Electrical and Computer Engineering
The University of Texas at Austin
Austin, TX, 78712-1084, USA
[email protected] 1, [email protected] 2, [email protected] 3
Abstract
Multiple-input-multiple-output (MIMO) communication systems use multiple antennas at both the
transmitter and the receiver. Under rich multipath environments with independent multipath fading
between each transmit and receive antenna pair, MIMO wireless communications systems achieve
significant capacity gains over conventional single antenna systems by exploiting the plurality of modes
present in the matrix channel within the same time-frequency slot. Moreover MIMO systems offer
significant diversity advantage over traditional wireless communication systems by exploiting both
transmit and receiver diversity by employing various space-time coding schemes. These have led to MIMO
being regarded as one of the most promising emerging wireless technologies. Our project deals with
MIMO based space-time coding and spatial multiplexing techniques that provide reliable capacity and
link gains, respectively. It is proposed to study, analyze and simulate the above-mentioned techniques. The
simulation results will allow us to compare performance of different space-time coding and spatial
multiplexing schemes.
I. Introduction
Physical limitations of the wireless medium provide a technical challenge for reliable wireless
communication. Techniques that improve spectral efficiency and overcome various channel impairments
such as signal fading and interference have made an enormous contribution to the growth of wireless
2
communications. Moreover, the need for high-speed wireless Internet has led to the demand for
technologies delivering higher capacities and link reliability than achieved by current systems. Multiple -
input multiple-output (MIMO) based communication systems are capable accomplishing these objectives.
MIMO systems are an extension of smart antennas systems. Traditional smart antenna systems
employ multiple antennas at the receiver, whereas in a general MIMO system multiple antennas are
employed both at the transmitter and the receiver. The addition of multiple antennas at the transmitter
combined with advanced signal processing algorithms at the transmitter and the receiver yields significant
advantage over traditional smart antenna systems—both in terms of capacity and diversity advantage.
A remarkable property of MIMO systems is their ability to turn multipath propagation into an
advantage. Indeed under rich multipath environments, with independently fading channels between every
pair of transmitter and receiver antennas, MIMO systems offer capacity increases that vary linearly with
the number of antennas used without increasing premium bandwidth or power [8]. Spatial multiplexing,
initially proposed by Paulraj and Kailath [25] and later practically demonstrated by Foschini [8] in Bell
Labs, is a technology that exploits this feature of MIMO systems to practically achieve the theoretical
capacity limit.
Apart from the capacity improvements mentioned above, MIMO systems can potentially offer
increased diversity advantage over traditional wireless systems. Under the assumption of sufficient
spacing between antenna elements and under rich multipath conditions, it can be shown that space-time
codes can be designed to provide diversity order (which is defined as the number of decorrelated spatial
branches) up toRT NN × , where
TN and RN are the number of transmit and receive antennas,
respectively [2].
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The above description gives but an idea of the potentials of MIMO systems. Indeed research in
this field is proceeding at a very rapid pace. For this reason there is a need for understanding the
fundamentals of MIMO communication systems. In this project we give a foundational introduction to
MIMO wireless communications systems. We implement important algorithms like spatial multiplexing
and space-time coding in a LabView environment and illustrate the advantages and limitations of these
different schemes. All the programs implemented in this project are available as a toolbox with a graphical
user interface (GUI) which can serve as a good educational tool for introducing MIMO systems to novices
in this area and students. Finally the copious references provided herein will direct the reader to the
frontiers of research in this exciting and rapidly growing field.
II. Multi-input Multi-output System Description and Channel Model
In the system design, we consider both discrete Rayleigh and Ricean flat fading MIMO channel
models, where TN and RN are the number of transmit and receive antennas respectively. In the
transmitter, a data stream is demultiplexed into TN independent substreams. Each substream is encoded
into transmit symbols using a modulation scheme (e.g. BPSK, QPSK, M-QAM, etc.) at symbol rate 1/T
symbol/sec with synchronized symbol timing [11].
First we describe the Rayleigh flat fading model. The baseband RN -dimensional received
signal vector TN krkrkrk
R)](),...,(),([)( 21=r at sampling instant k may be expressed as
)()()( kkk nxHr +⋅= (II-1)
where
=
RTR
T
NNN
N
hh
hh
LMOM
L
1
111H ,
=)(
)()(
1
kr
krk
RN
Mr ,
=)(
)()(
1
kx
kxk
TN
Mx ,
=)(
)()(
1
kn
knkn
RN
M .
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TN kxkxkxk
T)](),...,(),([)( 21=x denotes the transmit symbol vector with equally distributed transmit
power across the transmit power. Here, the superscript T is transposition. H denotes the RT NN ×
channel matrix, whose elements hmn at the mth row and nth column is the channel gain from the mth transmit
antenna to the nth receive antenna, and they are assumed to be independent and identically distributed
circularly symmetric complex Gaussian random variables with zero-mean and unit-variance, having
uniformly distributed phase and Rayleigh distributed magnitude. A circularly symmetric complex
Gaussian random variable is a random variable ),0(~ 2σC?jba + , in which a and b are i.i.d. real
Gaussian distributed as )2/,0( 2σ? . A commonly used channel model in MIMO wireless
communication systems is a block fading (also called quasi-static) channel model where the channel
matrix elements, which are independent and identically distributed (i.i.d.) complex Gaussian (Rayleigh
fading) random variables, are constant over a block and change independently from block to block. Hence,
we drop the index k for the channel gain. The elements of the additive noise vector
TN knknknkn
R)](),...,(),([)( 21= are assumed to be also white i.i.d. complex Gaussian random
variables with zero-mean and unit-variance. From this normalization of noise power and channel loss, the
averaged transmitted power which is equal to the average SNR at each receive antenna is to be no greater
than ? without regard to TN [11, 14, 27, 29].
Now we consider the Ricean fading model for the 2 transmitter and 2 receiver case which we use
in our simulations. The generalization to more than 2 transmit or receive antennas is straightforward. The
Ricean channel model is given by [18]
VF HK
HK
KH
11
1 ++
+= =
+
+
+ 2221
1211
11
1 2221
1211
hhhh
Keeee
KK
jj
jj
φφ
φφ
(II-2)
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where,
=
2221
1211
φφ
φφ
jj
jj
F eeee
H is a fixed matrix (i.e. fixed for each block of a block fading channel)
consisting of phase elements corresponding to the LOS component and VH is the Rayleigh flat-fading
matrix (as described before) i.e. the matrix element , jih denotes the Rayleigh flat fading channel
coefficient from transmitter i to transmitter j. K is the K-factor of the Ricean distribution which is
proportional to the strength of the line-of-sight (LOS) component.
The Ricean model (II-2) was suggested in [18] as a first order approximation to the Ricean
channel. We have decided to use this model in our simulations because under certain reasonable
assumptions it gives an accurate approximation to the Ricean channel [18]. The assumptions when using
this model are as follows: (i) that the distance between the antennas is sufficiently spaced, (ii) the
subscriber unit is mobile and possibly changing orientations with respect to the base station. Under these
assumptions it is reasonable that the phases of the LOS component arriving at the different receiver
antennas are random and uncorrelated.
In this project we study and simulate the performance of Space-Time coding and multiplexing
schemes. Figure II-1(a) and II-1(b) show a simplified system diagram of Space-time coding and V-BLAST
(Spatial multiplexing receiver) systems respectively. These will be considered in more detail in the
remaining of this paper.
(a) (b)
Figure II-1. System block diagram of (a) Space-time coding and (b) BLAST schemes [5, 11]
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III. Multi-input Multi-output System Channel Capacity
As mentioned already, MIMO systems provide significant capacity gain over single -input single -
output (SISO) channels. According to Shannon capacity of wireless channels, given a single channel
corrupted by an additive white Gaussian noise at a level of SNR, the capacity is
]1[log2 SNRC += [Bps/Hz] (III-1)
In the practical case of time-varying and randomly fading wireless channel, the capacity can be written as
]1[log 22 HSNRC ⋅+= [Bps/Hz] (III-2)
where H is the 11× unit-power complex Gaussian amplitude of the channel. Moreover, it has been
noticed that the capacity is very small due to fading events [9, 26, 29].
In the single -input multiple-output (SIMO) systems, the channel vector H is RN×1 and the
capacity is
]1[log2∗⋅+= HHSNRC [Bps/Hz] (III-3)
where the superscript * is the transpose conjugate. Compared with SISO system, the capacity of SIMO
system shows improvement. The increase in capacity is due to the spatial diversity which reduces fading
and SNR improvement. However, the SNR improvement is limited, since the SNR is increasing inside the
log function (III-4) [9, 26, 29].
For the MIMO system with TN transmit and RN receive antennas, it is shown that the
capacity is derived from
⋅+= ∗]det[log2 HHI
TN N
SNRC
R [Bps/Hz] (III-4)
Here, we can see that the advantage of MIMO systems is significant in capacity. As an example, for a
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large NNN RT == , NN IHH →∗ / , so the capacity is asymptotic to
]1[log 2 SNRNC +≈ [Bps/Hz] (III-5)
Therefore, the capacity increases linearly with the number of transmit antennas. Figure III-3 shows the
capacity for various multiple antenna systems. It is shown that SIMO systems offer smaller capacity gain
than MIMO systems. For example, MIMO system with 2== RT NN provides higher capacity than
SIMO system with 1=TN and 4=RN [9, 26, 27, 29].
Figure III-1. Capacity comparison of several multiple antenna systems
IV. Space-Time Coding Schemes
Space-time codes refer to the set of coding schemes that allow for the adjusting and optimization
of joint encoding across multiple transmit antennas in order to maximize the reliability of a wireless link.
Space-time coding schemes exploit spatial diversity in order to provide coding and diversity gains over an
uncoded wireless link. We first briefly the basics of diversity in the next section and thereafter we
elaborate on the prominent space-time coding schemes proposed in the literature.
IV.I. Overview of Diversity
Multipath fading is significant problem encountered in narrowband wireless communications.
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Fading results in fluctuations in the signal amplitude which degrades the BER performance at the receiver.
Diversity schemes attempt to mitigate this problem by finding independently faded paths in the mobile
radio channel. Usually this involves providing replicas of the transmitted signal over time, frequency or
space. Diversity is the single most important contributor to reliable wireless communications. Figure V-1
illustrates, by way of an example, the improvement in signal quality obtained when using diversity based
schemes. Notice how the deep fluctuations due to multipath fading are mitigated by making use of
independently faded copied arriving at different antennas. We shall now briefly review the 3 types of
diversity schemes that are exploited in wireless communications systems:
(i) Temporal Diversity: Here replicas of the transmitted waveform are provided across time by a
combination of channel coding and time interleaving strategies. The key for this form of diversity to be
useful is that the channel must provide sufficient variations in time. More precisely, this form of diversity
is useful when the coherence time of the channel is small compared to the desired interleaving symbol
duration (since otherwise time interleaving results in large delays).
Figure IV-1. Advantage of Diversity in Multipath Fading Condi tions
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(ii) Frequency Diversity: Here replicas of the transmitted waveform are provided in the form of
redundancies in the frequency domain. For this form of diversity to be useful the coherence bandwidth of
the channel must be small compared to the bandwidth of the signal. This criterion assures that different
parts of the relevant spectrum will suffer independent fades.
(iii) Spatial Diversity: Also known as antenna diversity, this is a very practical, effective and thus widely
used method for reducing the effect of multipath fading. Here replicas of the transmitted waveform are
provided across different antennas of the receiver. For this form of diversity to be effective, the antenna
spacing must be larger than the coherence distance in order to ensure independent fades across different
antennas. There are many traditional types of spatial diversity schemes [19] like selection diversity,
feedback diversity, maximal ratio receiver diversity (MRRD), and equal gain diversity. In the next section
we will find that space-time codes exploit a special type of spatial diversity which we refer to as space-
time diversity.
We emphasize again that the effectiveness of any diversity scheme lays the availability of
independently faded parts of the transmitted signal due to the random nature of the mobile radio channel
so that the probability of two or more relevant parts of the signal undergoing a deep fade is very small.
The constraints on coherence time, coherence bandwidth and coherence distance as stated above ensure
that this condition is met for the respective diversity schemes. The diversity system must then optimally
combine the received diversified waveforms so as to maximize the resulting signal quality. The reader can
refer to [19, 20] and references therein for a broad of overview of traditional diversity combining systems.
Diversity schemes can also be categorized in another way based on whether diversity is applied
at the transmitter or the receiver.
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(i) Receiver Diversity: Traditionally smart antenna systems employ receiver diversity schemes like
Maximum ratio combining [19] in order to improve the received signal quality. The major problem with
these schemes is the cost, size, and power in the remote units tending to be high. Receiver diversity is a
well studied subject [20]. But recently there has been much interest in another form of diversity called a
transmit diversity in the context of space-time coding.
(ii) Transmit Diversity: The idea behind these schemes is to introduce controlled redundancies at the
transmitter which can be exploited by appropriate signal processing techniques at the receiver. Note that
traditional channel coding can be considered a form of transmit diversity, however with the advent of
Space-Time block coding in [1] (which is a form of space time coding) this concept is extended to
encompass spatial diversity schemes. Space-time codes for MIMO in general exploit both transmit and
receiver diversity and as a result provide diversity advantage over traditional wireless systems (as the
latter only exploit receiver diversity). We will describe various space-time coding schemes in more detail
in the next section.
Another advantage of incorporating transmit diversity schemes in modern mobile communication
systems is that we can reduce the complexity of the remote devices and transfer it to the base station.
Portable hand-held mobile units are required to be inexpensive and low-complexity. Although economies
of scale will help in bringing down the cost of mobile units, increasing the complexity of the base station
(via adoption of space-time coding techniques) may be the only other possible trade space for meeting the
requirements of next generation communication systems [1].
IV.II. Space-Time Coding
Space-Time coding schemes primarily exploit a special form of spatial diversity which we refer
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to as space-time diversity. Unlike traditional spatial diversity schemes here information is encoded by
multiple transmitter antennas across both space and time. For the specific space-time codes considered
below we shall see how these codes, under certain conditions, fruitfully exploit this joint coding across
space and time to obtain diversity advantages as high RT NN × where TN is the number of transmit
antennas and RN is the number of receive antennas. Space-time coding can be broadly categorized into
2 classes
(i) Space-Time Block codes: These codes are transmitted using an orthogonal block structure which
enables simple decoding at the receiver.
(ii) Space-Time Trellis codes: These are convolutional codes extended to the case of multiple transmit
and receive antennas.
IV.II.I. Design Criteria for constructing Space-Time codes
In this section we derive optimality criteria which the specific Space-Time codes considered in
this paper are designed to meet. These optimality criteria were first proposed in [21] in the context of
space-time trellis codes. However these optimality criteria use metrics that can be used to evaluate the
performance of general space-time codes.
Consider a mobile communication system with NT transmit and NR receive antennas. The input
bit stream is coded by a space-time coding scheme and then split into NT bit streams. Each stream of data
is then pulse shaped, modulated and then launched into the wireless channel. The symbols TNiis 1= are
transmitted from the NT transmit antennas. All the NT symbols are transmitted in the same time slot and
subsequently mixed in the channel to obtain signal r(k) at the receiver antennas,
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)()()(..
.
.)(
11
kkHk
s
s
r
r
k
TR Nk
k
Nk
k
nsnHr +=+
=
= (IV-1)
where s(k) is the transmitted signal vector and n(k) is the noise vector.
Suppose that the code word c = TTT Nlll
NN ccccccccc ............ 212
22
121
21
11 is transmitted over l
consecutive time slots (with NT symbols simultaneously transmitted from the NT transmit antennas). The
probability that this code word is erroneously interpreted as e = Mlll
MM eeeeeeeee ............ 212
22
121
21
11 is given
by
))4/(),(exp(),...,2,1,,...,2,1,|( 02
, NEecdNjNihecP sRTji ≤==→ (IV-2)
where
2
1 1 1
2 )(),( ∑∑ ∑= = =
−=R TN
j
l
t
N
i
jt
itij cehecd is the distance between the corresponding received codewords.
This inequality (IV-1) follows by applying the applying the Chernoff bound to a Gaussian approximation
of the distribution of the distance between the received signals corresponding to codewords c and e .
The inequality (IV-1) above can be more compactly represented in terms the error matrix B(c, e)
corresponding to codewords c and e as follows. Let the error matrix B(c, e) be defined as follows
−−
−−
=
TTTT Nl
Nl
NN
ll
cece
cece
..........
..
11
1111
11
B (IV-3)
Then it follows that
∑∑ ∑ ∑== = =
ΩΩ=−=RR R N
jj
Hj
N
j
l
t
N
i
jt
itij cehecd
1
2
1 1 1
2 )(),( BB (IV-4)
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where ),...,,( ,,2,1 jNjjj Thhh=Ω .
Thus, ))4/(),(exp(),...,2,1,,...,2,1,|( 02
, NEecdNjNihecP sRTji ≤==→
∏=
ΩΩ−=RN
jsj
Hj NEececB
10
* )4/),(),(exp( B (IV-5)
Further we have that, ∑=
=ΩΩTN
ijiij
Hj ecec
1
2,
*),(),( βλBB
where TNii 1=λ are the eigenvalues of ),(),( ecec HBB obtained from diagonalization of ),(),( ecec HBB
==
M
H DVececV
λ
λ
0000.0000.0000
),(),(
1
*BB
where *
,1,2,1 ),...,,( Vjjjj Ω=βββ .
Ricean channel case
Now under the assumption of independent Ricean distributions of 2
,1 jβ we obtain (after
averaging the probability of error (IV-5) with the independent Ricean distributions of2
,1 jβ )
∏ ∏= =
+
−
+≤→
N
ji
s
is
jiM
i isji
NE
NE
K
NEhecP
1
0
0,
1 0, )
41
4exp(
))4/(1(1
)|(λ
λ
λ (IV-6)
where ),...,,( ,,2,1 jNjjj
TEhEhEhK = .
Rayleigh channel case
For the Rayleigh fading distribution case we obtain (by setting Ki,j = 0) that
RT
NN
i isji NE
hecP
+
≤→ ∏=1 0
, ))4/(1(1
)|(λ
(IV-7)
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Let r denote the rank of matrix HBB , then it follows that for the Rayleigh fading case:
TTR
TrN
sNr
ii
NN
i isji N
ENE
hecP−−
==
≤
+
≤→ ∏∏011 0
, 4))4/(1(1
)|( λλ
(IV-8)
The diversity gain and coding gain of a space-time coding scheme is given by TrN and TNr
ii
−
=
∏1
λ
respectively. The rank r of matrix HBB is also called the diversity-order of the space-time code.
Thus we arrive at the following design criteria for space-time codes for Rayleigh flat-fading
channel
(i) Rank criterion: To achieve maximum diversity of TR NN the matrix HBB must have full rank for
any pair of distinct codewords c and e .
(ii) Determinant criterion: For a given target diversity order nm the determinant of HBB must be
maximized in order to maximize the coding gain.
A similar design criterion also holds for the Ricean case [21]. The space-time codes we consider
in the following sections all meet the design criteria mentioned above.
We also infer from (IV-6) and (IV-7) that the worst case performance under a Ricean fading
condition will outperform the worst case performance under a Rayleigh fading condition in terms of BER
performance. Indeed it can also be shown [21] that even for dependent fade coeffic ients, any code that
works well for a Rayleigh channel cannot do worse under a Ricean fading condition (please see [21] for
more details) in terms of BER performance. The intuitive reason for this is clear—for the Ricean channel
the fades are not as deep as the Rayleigh fading case and so we can expect better BER performance for the
Ricean case.
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However, it is important to note that this gain in BER performance for Ricean channels comes at
the expense of capacity reduction because under correlated Ricean fading conditions MIMO systems can
no longer achieve significant capacity gains by exploiting the rich multipath structure of Rayleigh fading
channels.
IV.II.II. Space-Time Block Coding
The field of Space-Time Block coding was initiated by Alamouti [1]. The Alamouti codes
address the case of two transmit and NR receive antennas wherein it provides a diversity order of 2NR
which is equivalent to a NR branch Maximum Ratio Receiver Combining (MRRC) scheme. No knowledge
of channel state information (CSI) is assumed at the transmitter and perfect CSI is assumed at the receiver.
The MRRC scheme realizes spatial diversity gain by using multiple antennas at the receiver in
smart antenna system. We first briefly describe the MRRC scheme and then outline how the Alamouti
scheme performs similar to MRRC.
MRRC scheme
Figure IV-2 shows the channel model. Let the bit stream to be transmitted be indexed
as ,...],...,,[ 10 nsss . Let 0h be the channel between the first transmit antenna and the receiver and 1h be
the channel between the second transmit antenna and the receiver. Uncorrelated Rayleigh fading is
assumed between the transmitter and receiver 000
θα jeh = and 111
θα jeh = , where )1,0(, 10 CNhh ∈
are i.i.d. Complex Gaussian random variables. Thus the received signals at the 2 antennas
are 0000 nshr += and 1011 nshr += , where 10 ,nn are complex Gaussian noise. The receiver estimate
is formed by forming the following decision variable ,
1*10
*00
21
201
*10
*0 )|||(| nhnhshhrhrhy +++=+= (IV-9)
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So, if either 0h or 1h (or both) have not faded, then we can form an accurate estimate of the transmitter
by using a standard ML detector. Note that the probability that signals at both the channels will fade is
very low and so we can improve the average BER at the receiver output. We thus have a diversity of order
two.
Alamouti scheme
Figure IV-3 shows a schematic of the Alamouti scheme. This algorithm consists of three steps.
(i) Encoding and Transmission:
For a given symbol period, T, two symbols signals are transmitted simultaneously from the two
antennas. In the first symbol period the signal 0s is transmitted from the first antenna and signal 1s is
transmitted from the second antenna. During the next symbol period, signal )( *1s− is transmitted from
the first antenna and signal *0s is transmitted from the second antenna. In the encoding described above,
we observe blocks of symbols 0s and 1s are coded across space and time. For this reason Alamouti
codes are called Space-Time Block codes.
Figure IV-2. Two-Branch MRRC [1]
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It is assumed that the channel does not fade across these two symbol periods, that is,
00000 )()( θα jehTthth ==+= and 1
1111 )()( θα jehTthth ==+= .
Figure IV-3. Alamouti scheme [1]
Therefore the received symbols are given by 011000 )( nshshtrr ++== and
1*01
*101 )( nshshTtrr ++−=+= . Alternately we can represent the above equations in matrix form
nsHr += a (IV-10)
where
−
= *0
*1
10
hhhh
aH is the equivalent channel matrix , To ss ],[ 1=s is the transmitted block signal
vector, and To nn ],[ 1=n is the noise vector. Note that aH is an orthogonal matrix which is due to the
fact the transmitted symbol block has an orthogonal structure.
(ii) The combining step
Assuming perfect channel knowledge at the receiver, the vector r of received signal is left-
multiplied by HaH to obtain
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rHaHs
~= =
~
1
~
0
s
s = s
+
+2
12
0
21
20
0
0
hh
hh+
~n (IV-11)
where nHn~
Ha= .
Thus we have that
~
002
12
0*
110*0
~
0 )( nshhrhrhs ++=+= (IV-12)
~
112
12
0*
100*1
~
1 )( nshhrhrhs ++=−= (IV-13)
where
1*10
*0
~
0 nhnhn += and 1*10
*0
~
1 nhnhn −= (IV-14)
(iii) ML detection
The symbols ~
0s and ~
1s are sent to an ML Detector which amounts to minimizing the
following decision statistic
−++−−
2*01
*101
211000 shshrshshr over all possible values of 0s and
1s . Due to the orthogonality of the transmitted code blocks it can be easily shown that the above is
equivalent to minimizing the following decision statistic for detecting 0s
20
21
20
2
01*
1*00 ||)||||1( shhshrhr ++−+−+ (IV-15)
, and minimizing the following decision statistic for 1s
21
21
20
2
10*
1*10 ||)||||1( shhshrhr ++−+−− (IV-16)
This is a very simple decoding scheme involves decoupling of the signals transmitted from the different
antennas via linear combination at the receiver rather than joint-detection (which is done in space-time
trellis codes described in the next section).
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The Alamouti scheme can be similarly generalized to 2 transmit and RN receive antennas.
Further by the orthogonal property of the transmitted matrix it is easily shown in [3] that the Alamouti
code provides a maximal diversity advantage of TR NN =2 RN . The coding gain provided by this space-
time block coding scheme, which depends on the determinant criterion (mentioned in section IV.II.I), is
dependent on the particular modulation scheme used [3, 21]
The Alamouti is very attractive for practical applications because of the simplicity of decoding
combined with the maximal diversity gain advantage it provides.
Recent work
There has been work in the literature to extend Alamouti codes to the case of more than two
transmit antennas. However it turns out that in such cases it is not possible to design perfectly orthogonal
codes except in the case of real-valued modulations, like PAM. For complex constellation, fully
orthogonal codes cannot be constructed. The reader is referred to [3, 21, 22] for more details.
IV.II.III. Space-Time Trellis Coding
Space-Time trellis codes, which were originally introduced in [21], are convolutional codes
which are extended to the case of multiple transmit and receive antennas. No knowledge of channel state
information (CSI) is assumed at the transmitter and perfect CSI is assumed at the receiver. We define an r-
space-time code to be a space-time code that achieves a diversity order of r (which is the rank of matrix
HBB described in the section IV.II.I). The authors in [21] have proposed different trellis codes for TN =
2 transmitters, for 4-PSK and 8-PSK modulation schemes, and for different number of trellis states.
By analyzing the algebraic properties of these codes it can be shown that these codes satisfy the
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designed criteria derived in section IV.II.I. The space-time codes proposed in [21] satisfy the important
property of being geometrically uniform [21, 24]. From this property it immediately follows that the
proposed codes (like the one described in Figure IV-4) indeed maximize the coding gain for a given
diversity order. The reader is referred to [21, 24] for more details of the derivation.
Apart from maximizing the diversity and coding gain, we also would like to maximize, as much
as possible, the data rate of the transmission and to minimize the trellis complexity (which in turns
minimizes the complexity of the ML decoding algorithm). However there are tradeoffs in achieving these
conflicting goals. For example, diversity can be maximized by sending the same data across all the
different antennas but this would certain minimize the data rate of transmission. Our objective then must
be to maximize the data rate for a given diversity order. The authors in [21] explore fundamentals limits
underlying these tradeoffs and also furnish space-time coding schemes that achieve the above optimality
criterion. We briefly describe this in what follows.
Tradeoffs
We briefly describe some of the main theoretical results proved in [21] regarding the fundamental
limits on the rate, trellis complexity, diversity, constellation size, and their tradeoffs. We do not reproduce
the proofs these as the proofs can be found in [21] but we give some intuition and useful comments that
can allow for better appreciation of these results.
Theorem1: Consider a mobile communication system with n transmit, m receive antennas with a Ricean
fading model. Let rNR be the diversity advantage of this system. Assuming that the signal constellation Q
has 2b elements, the rate of the transmission R satisfies
21
l
rnAR
b l )],(log[2≤ (IV-17)
in bits per second per Hertz, where ),(2
rnA b l is the maximum size of a code of length n and minimum
Hamming distance r defined over an alphabet of size 2bl.
This theorem gives us an upper bound on the maximum achievable rate of any space-time code.
Later a constructive proof is given to show that the upper bound is achievable. From this theorem it is
immediately seen that to obtain a diversity advantage of RT NN , the transmission bit rate is at most b bits
per second per Hertz. For 4-PSK and 8-PSK constellations, for example, diversity advantage of RT NN
places an upper bound on the transmission rate of 2 and 3 bits/s/Hz respectively.
Another implication of Theorem 1, which is explored in the next two lemmas, is that there is a
fundamental tradeoff between constellation size, diversity, and the transmission rate.
Lemma 1: The constraint length of an r-space-time trellis code is at least r-1.
Lemma 2: Let b denote the transmission rate of a multiple antenna system employed in conjunction with
an r-space-time trellis code. The trellis complexity of the space-time code is at least 2b(r-1).
Lemma 2 shows how the trellis complexity is determined by the diversity r in the system and
maximum data rate b.
Code Construction
The authors in [21] construct codes that achieve the upper bound in Theorem 1 thus furnish a
constructive proof of the tightness of the upper bound. The constructive proof is given by demonstrating
codes that achieve this upper bound. Also, as mentioned before, these codes satisfy the design criteria
22
derived in section IV.II.I by virtue of being geometrically uniform. We consider one such space-time trellis
code below by way of an example. The interpretation and implementation of the other codes is very
similar.
Figure IV-4 shows the codes for a 2-space-time code using 4-PSK modulation with 4 states for
TN = 2 transmitters and date rate =2bps/Hz. The Trellis diagram is to be read as follows. The nodes
represent different state in the trellis diagram. The trellis diagram in Figure IV-5 consists of 4 states
(labeled states 0, 1, 2, 3). The symbols to the left of the each node correspond to the encoder state output
for the two transmit antennas (i.e. antenna 1, antenna2 read in this order) when excited by the
corresponding input bits as shown in top of each column. For example, when inputs bits 10 (respectively
for antennas 1 and 2) arrive in state 0, the encoder outputs a 0 on antenna 1 and a 2 on antenna 2. And,
following the state transition arrow corresponding to input 10, we see that the state changes to state 2. For
this 4-PSK scheme, the encoder outputs 0,1,2,3 are mapped to 1, j, -1, -j respectively.
Decoding Space-Time Trellis Codes
Given the received symbols r(k) in equation (IV-1) and assuming perfect CSI at the receiver, the
ML decoder is used to recover the input symbols as follows:
∑−
=
−=1
0
2minarg
TN
kkk
Sk Hsrs (IV-18)
where ],...,,[ 10 TNsssS = and minimization is done over all possible codeword matrices S.
We minimize over all possible codewords to obtain the ML estimate. The Viterbi algorithm is
used to compute the trellis path with the lowest metric. It is important that the complexity of ML decoding
for space-time codes grows exponentially with the number of number of constellations and trellis states.
For an r-space-time trellis coding scheme with a transmission data rate of b bits per second, the
23
complexity of the ML decoding algorithm is of the order )1(2 −rb .
Figure IV-4. Trellis diagram for n=2 with rate 2bps/Hz
IV.II.IV. Comparison of Space-time Block and Trellis codes
Space-time block codes cannot outperform the space-time trellis codes proposed in [21]. The
reason for this is that for space-time trellis codes the coding gain increases as we increase the number of
trellis states. So for a given diversity order trellis codes have more coding gain than space-time block
codes. A detailed proof of this can be found in [21].
Space-time block codes are computationally much more attractive than trellis codes. As
explained before, for Space-time block codes there exist simple ML decoding algorithms which only use
linear processing at the receiver by exploiting the orthogonality of the code design. On the other hand
Input Bits 00
01
10
11
State 0
Output for
Antenna1, Antenna2
00 01 02 03
State 1
Output for
Antenna1, Antenna2
10 11 12 13
State 2
Output for
Antenna1, Antenna2
20 21 22 23
State 3
Output for
Antenna1, Antenna2
30 31 32 32
State #
0
1
2
3
24
space-time trellis code requires computationally hungry trellis search algorithms which make it unsuitable
for most practical applications of MIMO systems.
IV.III. Simulations Results
The space-time coding schemes described above were simulated in LabView. Figure IV-5 shows
a comparison of BER performances of the Alamouti codes for 1, 2, and 3 receive antennas (and of course
for 2 transmit antennas). The x-axis shows the SNR in dB and the y-axis shows the BER. We see that we
get a significant improvement when using Alamouti codes over the case where we have no diversity,
especially for high SNR conditions. As we increase the number of receive antennas the BER performance
improves because the diversity order improves.
We simulated the performance of space-time trellis codes proposed in [21] for 4-PSK for
different number of transmit and receive antennas. Figure IV-6 shows the BER plot for 2 transmit and one
receive antenna for 4, 8 and 16 states. The number of states is associated with the trellis complexity of the
codes. Since the transmission rate (2 bps/Hz) is same for these three cases, by Lemma 2, the rank r
increases for increased trellis complexity which in turn decreases the probability of error as given in
Equation (IV-8). Thus as we increase the number of trellis states the BER decreases. But this comes at the
expense of increase complexity for the ML decoder at the receiver.
Figure IV-7 shows a comparison of BER performance for the case of 2 transmit and 2 receive
antennas with varying number of trellis states (4, 8 and 16 states). As expected the BER decreases as the
number of states increases.
Figure IV-8 shows a comparison of the trellis codes with Alamouti. Indeed, as explained earlier it
25
is guaranteed that Space-time block codes cannot outperform space-time trellis codes. However the
advantage of trellis codes is the simple decoding structure.
Figures IV-9 and IV-10 show the comparison between the performance of Rayleigh vs. Ricean
channel (K=0dB) for both Space-time block and trellis codes respectively. As explained in section IV.I.I.,
we expect an improvement in BER performance for the Ricean channel when compared to a Rayleigh
channel. This is indeed what we observe.
Figure IV-5. Performance of Alamouti Scheme
26
Figure IV-6. Performance of 2x1 Space-Time Trellis codes, 4PSK
Figure IV-7. Performance of 2x2 Space-Time Trellis codes, 4PSK
27
Figure IV-8. Comparison of Alamouti and Space-time trellis codes
Figure IV-9. Rayleigh vs. Ricean comparison for 2x2 Alamouti code, 4PSK
28
Figure IV-10. Rayleigh vs. Ricean comparison for 2x2 Space-time trellis code, 4PSK
29
V. Spatial Multiplexing Schemes
Spatial multiplexing scheme exploits the rich scattering wireless channel allowing the receiver
antennas to detect the different signals simultaneously transmitted by the transmit antennas. That is, spatial
multiplexing method uses multiple antennas at the transmitter and the receiver in conjunction with rich
scattering environment within the same frequency band to provide a linearly increasing capacity gain in
the number of antennas [9, 25, 26]. Hence, the concept of spatial multiplexing is different from that of
space-time coding method, which permits to efficiently introduce a space-time correlation among
transmitted signals to improve information protection and increase diversity gain. The basic spatial
multiplexing scheme is shown below.
Figure V-1. Basic Spatial Multiplexing Scheme with 3 transmit and 3 receive antennas [28]
Diagonal Bell Laboratories Layered Space-Time architecture (D-BLAST) is one of the spatial
multiplexing schemes to approach the theoretical capacity limit of multiple -input multiple-output (MIMO)
systems [9]. However, due to its complex coding procedure, Vertical Bell Laboratories Layered Space-
Time architecture (V-BLAST) has been proposed as a simplified version [10]. In V-BLAST, channel
coding may be applied to individual antennas (sub-layers), corresponding to the data stream transmitted
30
from each transmit antenna, while in D-BLAST coding processing is applied not only across the time but
also to each sub-layer, which implies higher complexity. The simple comparison between D-BLAST and
V-BLAST is shown at Figure V-1.
(a) (b)
Figure V-2. Transmit coding scheme comparison between (a) D-BLAST and (b) V-BLAST [31]
From the above figure, we can see that the essential difference between D-/V-BLAST is the
vector encoding process. In D-BLAST system, redundancy between the sub-streams is introduced by
using specialized inter-sub-stream block coding, and code blocks organized along diagonals in space-time
leads higher spectral efficiencies. On the other hand, in V-BLAST system, demultiplexing followed by
independent bit-to-symbol mapping of each sub-stream, so no coding is required [31]. From these
implemental advantages of V-BLAST, we will focus on the aspect of V-BLAST among several spatial
multiplexing methods.
V.I Receiver Design for Spatial Multiplexing System
The transmitted data stream of spatial multiplexing system is demultiplexed into TN lower rate
streams. After coding and modulation processing, each demultiplexed substream is transmitted
simultaneously from the TN transmit antennas. The substreams are co-channel signals, that is, they have
the same frequency band. At the receiver, each antenna observes a superposition of the transmitted signals,
31
separates them into constituent data streams, and multiplexed them in order to recover the original data
stream [26].
Receivers for spatial multiplexing can be divided into three classes; the maximum likelihood
receiver, the linear receiver, and the successive interference cancellation receiver.
A. Maximum likelihood (ML) Receiver
It is well known that theoretically, maximum likelihood detection algorithm is the optimum
method of recovering the transmitted signal at the receiver. ML receiver is a method that compares the
received signals with all possible transmitted signal vector which is modified by channel matrix H and
estimates transmit symbol vector x according to the Maximum Likelihood principle , which is shown as
2
,..., 1minargˆ k
TNkHxrx
xxx−=
∈ (V-1)
where x is the estimated symbol vector. The ML receiver searches through the all vector constellation
for the most probable transmitted signal vector. However, since the comparison complexity increases
exponentially with the number of transmit antennas, it is very difficult to use the receiver in practice,
which is the main disadvantage of this method. For example, in the case of 4 transmit antennas and 16-
QAM transmission, a total of 65536164 = caparisons per symbol is required to be enumerated for each
transmitted symbol. Therefore, the complexity of ML receiver is high and even prohibitive when many
antennas or high order modulation schemes are used [17, 27, 28].
B. Zero Forcing (ZF) Receiver
Zero-forcing receiver is a simple linear receiver, which uses a straight matrix inversion. Under
the assumption that the channel matrix H is invertible, a transmitted data symbol vector can be estimated
as
32
xHHxHHx * +− == 1)(ˆ (V-2)
where + represents the pseudo-inverse [29]. Note that the inverse only exists if the columns of H are
independent, since it is assumed that the elements of H are i.i.d. From the above equation, we can see that
the ZF receiver can separates the co-channel signals ,..., 1 TNk xxx ∈ . On the other hand, ZF receiver
results in a poor performance in the low SNR regime, when the channel matrix H becomes very ill-
conditioned in certain fading events [17, 27, 29].
C. Minimum Mean-Square Error (MMSE) Receiver
Another linear detection algorithm to the problem of estimating a random vector x on the basis of
observations y is to choose a matrix D that minimizes the Mean Square Error [27]
)][()]ˆˆ[(2 Dy(xDy)xx(x)xx −−=−−= ∗∗Eε (V-3)
The solution of the linear MMSE is given by
rHHHIrDx ⋅+=⋅= − HHNRSNR
1)1
(ˆ (V-4)
where the superscript H denotes the complex conjugate transpose [14]. The ZF receiver perfectly separates
the co-channel signals at the cost of noise enhancement. On the other hand, the minimum mean-square
error (MMSE) receiver can minimize the overall error caused by noise and mutual interference between
the co-channel signals, which is also at the cost of reduced signal separation quality [17, 27].
D. Successive Interference Cancellation (SIC) Receiver
The Successive Interference Cancellation (SIC) algorithm is in general combined with V-BLAST
receiver. Therefore, the SIC receiver is called just as V-BLAST receiver, which is an attractive alternative
to ZF and MMSE receivers. The SIC receiver provides improved performance at the cost of increased
computational complexity. Rather than just jointly decoding the transmitted signals, this nonlinear
33
detection scheme first detects the strongest transmitted signal, cancels the effect of this strongest signal
from each of the received signa ls, and then proceeds to detect the strongest of the remaining transmitted
signals, and so on. Under the assumption that the channel matrix H is known, the basic steps of the SIC
algorithm is summarized as follows [27, 32]
Ordering: Determine the optimal detection order by choosing the row with minimum
Euclidian norm
Nulling: Estimate the strongest transmit signal by nulling out all weaker transmit signals
Slicing: Detect the value of the strongest transmit signal by slicing to the nearest signal
constellation value
Cancellation: Cancel the effect of the detected strongest transmit signal form the
received signal vector in order to reduce the detection complexity for remaining transmit signals
Since the SIC algorithm is combined with ZF receiver or MMSE receiver, it can accordingly be
classified into the ZF V-BLAST receiver and the MMSE V-BLAST receiver.
D-1 Zero Forcing V-BLAST (ZF V-BLAST) Receiver
The Zero Forcing V-BLAST receiver algorithm has been presented in [10]. First, we let the
ordered set ,..., 21 TNkkkS = be a detection order of sub-streams. According to the basic SIC algorithm,
the ZF V-BLAST detection algorithm can be described as a recursive procedure, including determination
of the optimal ordering, as follows [10, 11]
¨ Initialization
211
1
1
)(
1
minarg jj
k
i
G
HG
rr
=
=
=←
+
34
¨ Recursion
As we assumed already, all components of x utilize the same constellation. Therefore, iky with
the lowest post-detection SNR will dominate the error performance of the detection algorithm. Due to
symbol constellation, these post-detection SNRs depend on the order in which the decision statistics are
computed. Thus, an obvious aspect of this system is the maximization of the worst of these post-detection
SNRs. The local optimization of choosing the component with the best SNR at each stage, leads to the
globally optimal ordered set S in this maximum sense. In [10], they proved this optimal ordered set
concept. Since the symbolikx is assumed to have the unit average power, the post-detection SNR
ikρ
for the ik th symbol is calculated as follow
(V-5)
D-2 Minimum Mean-Square Error V-BLAST (MMSE V-BLAST) Receiver
MMSE V-BLAST receiver suppresses both the interference and noise components, whereas ZF
V-BLAST receiver removes only the interference components. That is, the mean square error between the
transmitted symbols and estimate of the receiver is minimized. Thus, MMSE V-BLAST receiver yields
superior performance to ZF V-BLAST in the presence of noise [33, 34]. The detection algorithm is
( )
1
)(ˆ
)(ˆ
)(
21
,,1
1
1
T
minarg1
1
+←
=
=
−=
=
=
=
+∉
+
++
+
ii
k
a
yQa
y
jikkj
i
ki
kkii
kk
ikk
kik
i
i
ii
ii
ii
i
G
HG
Hrr
rw
Gw
L
22
2
i
i
i
k
k
k
a
wσρ =
35
summarized as follows
¨ Initialization
¨ Recursion
where 2σ is the variance of i.i.d complex Gaussian random noise with zero mean. Furthermore, the
detection ordering is determined based on the SINR. As the similar way of ZF V-BLAST, the post-
detection SINR ikρ for the ik th symbol is calculated as follow [34]
(V-6)
V.II Simulation Results
We simulated the average symbol error rate (SER) with 0/ NEs for various antenna
configurations with the modulation of 4-QAM constellation, which is the same as QPSK that we
simulated at the space-time coding schemes. Furthermore, we simulated the several spatial multiplexing
schemes on the Rayleigh and Ricean fading channels for the comparison of the difference of the
( )j
jk
i
)SINR(
1
11
H12H1
1
maxarg=
+=
=←
−HIHHG
rr
σ
( )
( )
1
SINR
)(ˆ
)(ˆ
)(
1,,
1
H12H1
1
T
maxarg1
1
+←
=
+=
−=
=
=
=
+∉
+
−+
+
ii
k
a
yQa
y
jikkj
i
iiii
kkii
kk
ikk
kik
i
ii
ii
ii
i
L
HIHHG
Hrr
rw
Gw
σ
( )( ) 2
11
22
2
1
1
1
∑≠
−
−
+=
iiii
iii
i
klkkk
kkk
kw Hw
Hw
σρ
36
performances.
In Figure V-3, we provide the SERs for the different antenna configuration of 3,2,1=TN
and 3,2,1=RN against 0/ NEs . It can be seen that the diversity order of a system based on the ML
technique is equal to the number of receiver antennas RN , therefore, the SER performance increases by
introducing an extra transmit and an extra receive antenna. However, it should be noted that the SER
performance of a ML receiver system does not lose its diversity order if the number of transmit antennas
increased, but the overall performance deteriorates. For example, we can see the curves of 1×2 and 2×2
antenna configurations. That is, the curves of Figure V-3 shift to the right when more transmit antennas are
added. The reason for this degradation is the increase in the error space with increasing number of
antennas. As the number of transmit antennas is increased, the size of the input vector constellation
increases. Thus, the number of signal vector which can be mistaken for is more, which implies an increase
in the number of possible error vector resulting in performance degradation [17].
Figure V-3. Symbol Error Rate (SER) comparison for various antenna configurations over Spatial
multiplexing scheme by using Maximum Likelihood (ML) receiver technique
37
Figure V-4 shows the SER performance when ZF technique is used at the receiver, which also
use 4-QAM transmit constellation on all antennas. When comparing it with ML receiver system and SISO
system, it is seen that SISO with 1,1 == RT NN , ML receiver with 1,1 == RT NN and MIMO with
2,2 == RT NN show almost the same SER performance representing Rayleigh flat fading. The reason is
that when RT NN = , the 0/ NEs on each stream is chi-squared variable with two degrees of freedom
which is a Rayleigh flat fading. Thus, there is no diversity benefit from the multiple receive antennas [17].
However, when RT NN < , the ZF receiver technique shows the SER performance increase, as shown in
Figure V-3.
Figure V-4. Symbol Error Rate (SER) comparison for various antenna configurations over Spatial
Multiplexing scheme by using Zero Forcing (ZF) receiver technique
38
Figure V-5. Symbol Error Rate (SER) comparison for various antenna configurations over spatial
multiplexing scheme by using Minimum Mean-Square Error (MMSE) receiver technique
In Figure V-5, the SER performance comparison is depicted when MMSE technique is used. The
trend of SER curves is very similar to that of ZF receiver technique. As we already discussed, the MMSE
receiver minimizes the overall error caused by noise and mutual interference between the co-channel
signals, thus it is less sensitive to the channel noise at the cost of reduced signal separation [27]. Therefore,
although the MMSE receiver show a little better performance than ZF receiver, in the high SNR case the
MMSE receiver converges to the ZF receiver [17].
The next two Figures V-6 and V-7 show the SER vs. 0/ NEs of ZF V-BLAST and MMSE V-
BLAST, respectively. As compared to the above V-4 and V-5, these two figures indicate that the ZF and
MMSE receivers with Successive Interference Cancellation (SIC) method present better performance.
Especially for ZF V-BLAST, the improvement is approximately 3dB at the target SER of 10-4 with the
antenna configuration of 2 transmit and 3 receive antennas as shown in Figure V-7. Although the
39
improvement of ZF V-BLAST is not better than that of ML receiver, ZF V-BLAST has less complexity
compared to ML receiver. When we apply MMSE V-BLAST, the improvement compared to only MMSE
scheme is about 3 dB at the target SER of 10-4 with the 2 transmit and 4 receive antenna configuration. As
we already studied, we can see that the improvement is due to the optimal ordering combined with the
successive cancellation of the interference and noise. All of the mentioned spatial multiplexing receivers
are compared in Figure V-8. From the Figure V-8, we can see that the MMSE V-BLAST and ZF V-BLAST
are intermediate and applicable in terms of performance and complexity.
Figure V-6. SER comparison for various antenna configurations over spatial multiplexing scheme by using
Zero Forcing (ZF) receiver with successive interference cancellation receiver technique (ZF V-BLAST)
40
Figure V-7. SER comparison for various antenna configurations over spatial multiplexing scheme by using
Minimum Mean-Square Error (MMSE) receiver with successive interference cancellation technique (MMSE
V-BLAST)
Figure V-8. Comparison of ML, ZF, MMSE, ZF V-BLAST and MMSE V-BLAST receivers with SER
vs. 0/ NEs for various antenna configurations over spatial multiplexing scheme
Finally, the next five figures show the comparison of the spatial multiplexing schemes applied to
Rayleigh and Ricean fading channels. Under the assumption that the all transmit and receive antennas are
41
uncorrelated with each other, the SERs of the spatial multiplexing methods in the Ricean channel indicate
the better performance.
Figure V-9. Rayleigh vs. Ricean comparison for 2x2 ML receiver, 4QAM
Figure V-10. Rayleigh vs. Ricean comparison for 2x2 ZF receiver, 4QAM
42
Figure V-11. Rayleigh vs. Ricean comparison for 2x2 MMSE receiver, 4QAM
Figure V-12. Rayleigh vs. Ricean comparison for 2x2 ZF V-BLAST receiver, 4QAM
43
Figure V-13. Rayleigh vs. Ricean comparison for 2x2 MMSE V-BLAST receiver, 4QAM
44
VI. Conclusions
The increasing demand for the development of wireless communication systems for high data
rate transmission and high quality information exchange leads to the new challenging subjects in the
telecommunication research area. Therefore, The reason that we chose ‘Fundamental overview and
simulation of MIMO systems for Space-Time coding and Spatial Multiplexing’ as a semester project for
EE381k-11 Wireless communication is because of the fact that the MIMO principle is able to provide
future wireless communication systems with significant increased capacity or higher link reliability using
the same bandwidth and transmit power as today.
This project investigated the information theoretical background for MIMO wireless
communication systems. The MIMO capacity formulae were simply explained, which showed the
dramatically increased capacity and reliability as compared with conventional single -input single-output
systems. The basic idea of MIMO system architecture is to usefully exploit the multipath, rather than
mitigate it, considering the multipath itself as a source of diversity which allows the parallel transmission
of a certain number of substreams. The indoor wireless environment is an good example of a rich
scattering environment necessary to obtain the best performance promised by MIMO systems. Several
Space-Time Coding (STC) schemes such as Space-Time Block Coding (STBC) and Space-Time Trellis
Coding (STTC), and Spatial Multiplexing schemes such as V-BLAST exploit the rich scattering wireless
channel allowing the receiver to detect the different signals simultaneously transmitted by the multiple
transmit antennas. STC permits to efficiently introduce a space-time correlation among transmitted signals
that can be exploited to increase information reliability while spatial multiplexing is to increase data rate.
Simulations and measurements were presented to show the capability for several MIMO systems
(STC and Spatial Multiplexing schemes). It could be shown that the reliability and capacity significantly
45
increase with increasing the number of transmit and receive antennas. STBC can realize full diversity gain
and especially, among a number of antenna diversity methods, the Alamouti method is very simple to
implement. Hence, we derived and simulated the performance of the Alamouti coding method and varied
the number of transmit and receive antennas. It was shown that STTC provides better performance than
STBC at the cost of increased receiver complexity. Moreover, we theoretically investigated spatial
multiplexing methods, and then evaluated and simulated the performance V-BLAST, which exhibit the
best trade-off between performance and complexity among spatial multiplexing techniques. In addition,
we derived its several receiver structures such as Maximum Likelihood (ML), Zero-Forcing (ZF),
Minimum Mean-Square Error (MMSE), and Successive Interference Cancellation (SIC) Receivers. By
investigating and simulating each receiver concepts, it was shown that V-BLAST implements a detection
technique, i.e. SIC receiver, based on ZF or MMSE combined with symbol cancellation and optimal
ordering to improve the performance, although ML receiver appears to have the best SER performance.
In this project, the MIMO principle is based on a rich multipath environment without a normal
Line-of-Sight (LOS), that is, the Rayleigh flat fading channel. However, due to movement or other
changes in the environment, LOS situation can arise. Thus, the Ricean channel modeling for MIMO
systems has recently attracted considerable attention and been researched [35, 36]. Therefore, we also
simulated the MIMO system operations in the Ricean channel and achieved the result that the MIMO
systems in a Ricean channel show the better performance than in a Rayleigh channel under the assumption
of uncorrelated antennas. For the case of correlated antennas, future works are necessary.
On the other hand, in order to achieve high gains for a Rayleigh frequency selective fading
channel, the employment of an orthogonal frequency division multiplexing (OFDM) communication
system in MIMO systems has also been actively researched, which is called MIMO-OFDM system [37-
46
40]. It is from the fact that the OFDM system has simple implementation of equalizers and robustness
against frequency selective channels by converting the channel into the flat fading subchannels, that is, a
broadband signal is broken into multiple narrowband tones, in which each carrier is more robust to
multipath [39, 40]. Moreover, space-time coded OFDM (STC-OFDM) and BLAST-OFDM have proposed
to overcome frequency selective fading channels and achieve higher capacity and spectral efficiency in a
rich scattering environment [41-45].
From our study and overview of MIMO systems, it is clear that MIMO systems offer significant
gains in performance over traditional wireless communication systems. For this reason, MIMO technology
is poised to play an important role in the next generation mobile communication systems and standards.
MIMO technology has already been incorporated into wireless LAN standards such as IEEE 802.11 and
HiperLAN/2 and is an optional feature of the 3G UTMS (IMT-2000), a next generation wireless
communications standard. Furthermore, MIMO technology is a strong candidate for 4G along with OFDM.
47
References
[1] Alamouti, S. M., “A simple transmit diversity technique for wireless communications,” Selected
Areas in Communications, IEEE Journal,16(8):1451–1458, 1998
[2] Tarokh, V., Naguib, A., Seshadri, N., and Calderbank, A.R., “Space-time codes for high data rate
wireless communication: performance criteria in the presence of channel estimation errors,
mobility, and multiple paths,” Communications, IEEE Transactions, Volume: 47 Issue: 2 , pp.
199-207, Feb 1999
[3] Tarokh, V., Jafarkhani, H., and Calderbank, A. R., “Space-time block codes from orthogonal
designs,” IEEE Trans, Inform. Theory, Vol. 45, No. 5, pp. 1456-1467, July 1999
[4] Tarokh, V., Naguib, A., and Calderbank, A. R., “Combined array processing and space-time
coding,” IEEE Trans, Inform. Theory, Vol. 45, No. 4, pp. 1121-1128, May 1999
[5] Tarokh, V., Jafarkhani, H. and Calderbank, A. R., “Space-time block coding for wireless
communications: performance results,” IEEE Journal of selected areas in Communications, Vol.
17, No. 3, pp. 451-460, March 1999
[6] Sandhu, S. and Paulraj, A., “Space-time block codes: a capacity perspective”, IEEE
Communications Letters, Volume: 4 Issue: 12, pp. 384-386, Dec 2000
[7] Telatar, I. E., “Capacity of multi-antenna Gaussian channels,” Tech. Rep. #BL0112170-950615-
07TM, AT&T Bell Laboratories, 1995
[8] Foschini, G. J. and Gans, M. J., “ On limits of wireless communications in a fading environment
when using multiple antennas,” Wireless Personal Communications, vol. 6, pp. 311-335, 1998
[9] Foschini, G. J., “Layered space-time architecture wireless communication in a fading
environment when using multi-element antenna,” Bell Labs Tech. J., pp. 41-59, Autumn 1996
[10] Wolniansky, P. W., et al, “V-BLAST: An architecture for realizing very high data rates over the
rich-scattering wireless channel,” Proc. IEEE ISSSE-1998, Pisa, Italy, 30 September 1998
[11] Golden, G. D., Foschini, C. J., Valenzuela, R. A., and Wolniansky, P. W., “ Detection algorithm
48
and initial laboratory results using V-BLAST space-time communication architecture,” IEE Lett.,
Vol. 35, No. 1, pp. 14-16, January 1999
[12] Burr, A.G. , “Application of space-time coding techniques in third generation systems,” 3G
Mobile Communication Technologies, 2000. First International Conference on (IEE Conf. Publ.
No. 471), 2000
[13] Hochwald, B. M. and Marzetta, T. L., “Unitary space-time modulation for multiple -antenna
communications in Rayleigh flat fading,” IEEE Trans. Inf. Theory, Vol. 46, No. 2, pp. 543-564,
2000
[14] Baro, S. Bauch, G, Pavlic, A., and Semmler, A., “Improving BLAST performance using space-
time block codes and turbo decoding,” IEEE Global Telecommunications Conference, 2000
[15] http://www.ece.utexas.edu/~wireless
[16] Al-Dhahir, N., Fragouli, C., Stamoulis, A., Younis, W., and Calderbank, R., “Space-time
processing for broadband wireless access,” IEEE Communications Magazine, Volume: 40, Issue:
9, pp. 136-142, 2002
[17] Gore, D. A., Heath, R. W. Jr., and Paulraj, A. J., “Performance Analysis of Spatial Multiplexing
in Correlated Channels,” submitted to Communications, IEEE Transactions March 2002.
[18] Erceg, V., Soma, P., Baum, D.S., Paulraj, A.J., “Capacity Obtained from Multi-Input-Multi-
Output Channel Measurements in fixed Wireless Environments at 2.5GHz,” Communications,
2002. ICC 2002. IEEE International Conference on , Volume: 1 , 2002, Page(s): 396 –400
[19] Rappaport, T. S., “Wireless Communications: Principles and Practice,” Prentice Hall PTR, 2000
[20] Jakes , W. C., “Microwave mobile communications,” New York: Wiley, 1974
[21] Tarokh, V., Jafarkhani, H., and Calderbank, A. R., “Space-time Codes for High Data Rate
Wireless Communication: Performance Criterion and Code Construction,” IEEE Trans. Inform.
Theory, Vol. 44, No. 2, pp. 744-765, July 1998
[22] Jafarkhani, H., “A quasi orthogonal space time block code block code,” IEEE Trans. Comm., vol.
49
49, pp. 1-4, Jan 2001
[23] Tirkkonen, O., Boariu, A., and Hottinen, A., “Minimal non orthogonality rate 1 space-time block
code for 3+ tx antennas,” Proc. IEEE Int. Symp. Spread Spectrum Technology, 2000
[24] Forney Jr., G.D., “Geometrically uniform codes,” IEEE Trans, Inform. Theory, Vol. 37, No. 5, pp.
1241-1260, September 1991
[25] Paulraj, A. and Kailath, T. , “Increasing capacity in wireless broadcast systems using distributed
transmission/directional reception,” U. S. Patent, no. 5,345,599, 1994
[26] Gesbert, D., Bolcskei, H., Gore, D., and Pauraj, A., “MIMO wireless channels: capacity and
performance prediction,” Global Telecommunications Conference, 2000. GLOBECOM`00.
IEEE, Vol. 2, pp. 1083-1088, 2000
[27] Bolcskei, H. and Paulraj, A., “Multiple -Input Multiple-output (MIMO) wireless systems,” In J.
Gibson, editor, The communications Hadbook. CRC Press, 2001
[28] Gesbert, D. and Akhtar, J., “Breaking the barriers of Shannon's capacity: An overview of MIMO
wireless system,” Telektronikk Telenor Journal. Jan. 2002
[29] Zelst, A. van, “ Space Division Multiplexing Algorithm,” Proc. of IEEE MEleCon 2000.3, pp.
1218-1221, May 2000
[30] Kailath, T., Sayed, A. H., and Hassibi, B., “State Space Estimation,” Prentice-Hall, New Jersey,
1999
[31] www.hut.fi/Units/Radio/courses/S26300/3A.pdf
[32] Guo, Z. and Nilsson, P., “On Detection Algorithms and Hardware Implementations for V-
BLAST,” SSoCC’02, Faukenberg, 2002
[33] Tidestav, C., Ahlen, A., and Sternad, M., “Realizable MIMO decision feedback equalizers:
structure and design,” IEEE Trans, Signal Processing, Vol. 49, pp. 121-133, Jan. 2001
[34] Debbah, M., Muquet, B., de Courville, M, Muck, M., Simoens, S., and Loubaton, P., “ MMSE
50
Successive Interference Cancellation Scheme for new Spread OFDM Systems,” Vehicular
Technology Conference, Tokyo, Japan, May 2000
[35] Moustakas, A. L., et al., “ Communication through a diffusive medium: Coherence and
Capacity,” Science, Vol. 287, pp. 287-290, Jan. 2000
[36] Shiu, D. S., et al., “Fading Correlation and its effect on the capacity of multielement antenna
systems,” IEEE Trans. Commun., Vol. 48, pp. 502-513, Mar. 2000
[37] Raleigh, G., and Cioffi, J. M., “Spatio-temporal coding for wireless communication,” IEEE
Trans. on Commun., COM-46, pp. 357-366, Mar. 1998
[38] Chaufray, J. M., et al., “Consistent estimation of Rayleigh fading channel second-order statistics
in the context of the wideband CDMA mode of the UMTS,” IEEE Trans. Signal Processing, Vol.
49, No. 12, pp. 3055-3064, Dec. 2001
[39] Barhumi, I., Leus, G. , and Moonen, M., “Optimal training sequences for channel estimation in
MIMO-OFDM systems in mobile wireless channels,” Broadband Communication, 2002. Access,
Transmission, Networking. 2002, International Zurich Seminar on, pp.44-1-6, 2002
[40] Sampath, H., Talwar, S., Tellado, J, Erceg, V., and Paulraj, A., “A Fourth-Generation MIMO-
OFDM broadband wireless system: Design, Performance, and Field Trial Results,” IEEE
Communications Magazine, Vol. 40, Issue:9, pp. 149-149, 2002
[41] Lu, B. and Wang, X., “Space-time code design in OFDM systems,” Global Telecommunication
Conference, 2000. GLOBECOM`00. IEEE, Vol. 2, pp. 1000-1004, 2000
[42] Piechocki, R., Fletcher, P., Nix, A., Canagarajah, N., and McGeehan, J., “Performance of space-
time coding with HIPERLAN/2 and IEEE 802.11a WLAN standards on real channels,”
Vehicular Technology Conference, 2001, VTC 2001 Fall. IEEE VTS 54th, Vol. 2, pp. 848-852,
2001
[43] So, D. K. C, and Cheng, R. S., “ Performance evaluation of space-time coding over frequency
selective fading channel,” Vehicular Technology Conference, 2002, VTC 2002 Spring. IEEE
VTS 55th, Vol. 2, pp. 635-639, 2002
51
[44] Wang, L., Ren, Y., and Shan., X., “Effect of carrier frequency offset on performance of BLAST-
OFDM systems,” Electronics Letters, Vol. 38, Issue:14, pp. 747-748, Jul. 2002
[45] Piechocki, R. J., Fletcher, P. N., Nix, A.R. Canagarajah, C. N., and Mcgeehan, J. P.,
“Performance evaluation of BLAST-OFDM enhanced Hiperlan/2 using simulated and measured
channel data,” Electronics Letters, Vol. 37, Issue:18, pp. 1137-1139, Aug. 2001