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Proceedings ArticleDOI

Adaptive Receiver Beamforming for Diversity Coded OFDM Systems: Maximum SNR Design

01 Nov 2005-pp 1-5

TL;DR: A new frequency-time encoding scheme is introduced that can be used to exploit frequency diversity branches for broadband OFDM systems with only one antenna at the MS and is shown to have a 4-dB improvement over the conventional space-time coding scheme when two receive antennas are used.
Abstract: Over the recent years, advance multiantenna transmission schemes have attracted considerable interest due to their potential benefits in improving the system capacity and error-rate performance. As a result, space-time coding and transmit beamforming have emerged as the two most promising techniques. Because of limited space at the mobile station (MS) and the fact that download intensive services are to be introduced in the next generation of cellular systems, most of research efforts have been pouring on transmit diversity techniques. In this paper, we focus on adaptive uplink transmission and reception techniques for wireless communications and introduce a new frequency-time encoding scheme that can be used to exploit frequency diversity branches for broadband OFDM systems with only one antenna at the MS. By incorporating this with receive beamforming at the base station (BS), the instantaneous signal- to-noise ratio (SNR) is maximized and the system error-rate performance is then further improved. Numerical results showed that systems employed the proposed transceiver structure have a 4-dB improvement over the conventional space-time coding scheme when two receive antennas are used.
Topics: Transmit diversity (61%), Diversity scheme (58%), Beamforming (58%), Space–time code (55%), Base station (54%)

Summary (3 min read)

Introduction

  • I. INTRODUCTION Signal transmission in multi-input multi-output (MIMO) systems that employs more than one antennas at the transmitter and the receiver has shown to be effective in exploiting spatial diversified paths of wireless channels [1]-[2] and increasing both system capacity and error-rate performance.
  • Their performance improvements are based on the assumption that the arriving multipath signals are sufficiently uncorrelated.
  • The application of space-time coding to orthogonal frequency division multiplexing (OFDM) systems was first introduced in [8].
  • With the effective use of beamforming at the multiantennas base station (BS) receiver, the optimal adaptive beam-mapping weights is applied to maximize the instantaneous SNR, and thus, system error-rate performance during uplink transmission in a singleinput multi-output (SIMO) channel is further enhanced.
  • ‖ · ‖F is the Frobenius norm; √A stands for Hermitian square root of matrix A; det(·) denotes the determinant; E{·} is the expectation operator.

II. SYSTEM MODEL

  • It is assumed that the channel coherent bandwidth is larger than the bandwidth of each subcharrier; the authors thus consider the corresponding subchannel to be frequency-flat.
  • The normalized correlation matrix that specifies the correlation between antenna elements is Authorized licensed use limited to: RMIT University.
  • Let us denote the correlated SIMO channel frequency response vector for the kth subcarrier as hk ∈ C1×Nr .
  • ]T is the corresponding discrete Fourier transform coefficients.
  • Furthermore, quasi-static fading is also assumed throughout the duration of one FTBC codeword length but fading may vary from one block to another.

A. Subchannel Grouping & Frequency-Time Encoding

  • The concept of subchannel grouping, sometimes referred as subcarrier grouping, was originally used in adaptive modulation scheme in [10] to reduce processing complexity by grouping subcarriers or subchannels that are within one channel coherent bandwidth and having a similar fading gain.
  • The authors utilize the concept of subchannel grouping in [12] by treating subcarriers that are having different fading gains as additional antennas at the MS.
  • By doing so, the authors can then directly apply the spacetime codes in [3]-[5], [15] in their system by spreading symbol energy across OFDM frequencies instead of antennas.
  • If the well-known Alamouti’s space-time block code in [3].
  • Note that other STBC matrices in [4] and STTC technique in [5] can be applied in the same way, but with different subchannel grouping sizes.

B. Receiver Beamforming

  • Assume ideal symbol-time sampling and carrier synchronization, the discrete time baseband equivalent form of system input-output equation can be expressed as rk = xkhk √.
  • Rr + ek, (2) where xk is an FT encoded data symbol transmitted on the kth subcarrier, and ek is a additive white Gaussian noise vector with each element having zero mean and σ2k variance.
  • Authorized licensed use limited to: RMIT University.
  • Downloaded on November 19, 2008 at 19:08 from IEEE Xplore.
  • The antenna weight mapping process is performed across.

C. Frequency-Time Decoding

  • To recover original data, channel estimation results are also used for the FT decoding process.
  • Since frequency subcarriers are treated as additional transmit antennas at the transmitter FT encoding, the decoding algorithm is divided into a combining stage and a maximum likelihood decoding (MLD) stage.
  • The combining stage is simply adding the received signals in different subcarriers that are within the same subchannel group.
  • Denote i(m) as the frequency index for subchannel group i.
  • Thus, without the presumption that Alamouti’s space-time block code is used for the FT encoding, the combining process can be expressed as yi(n) = Nc/Ng∑ m=1 yi(m)(n).

D. Signal-to-Noise Ratio (SNR)

  • Given that the average transmitted energy during one OFDM-symbol interval is E{xk} = E{|sn|2} = εs , (7) the received SNR at the kth subcarrier for the detection of xk has the form: γk = εs‖hk √ RrWk‖2F σ2k‖Wk‖2F . (8) Denote |sn − s̃n| as the minimum distance between the underlying constellation symbols, then the symbol energy for both QAM and PSK modulation schemes are given in [16] as εs = (M − 1)|sn − s̃n|2QAM 6 , for QAM (9) εs = |sn − s̃n|2PSK 4 sin2(π/M) , for PSK. (10).

III. OPTIMAL ANTENNA WEIGHTING MATRIX: MAXIMUM SNR DESIGN

  • The objective of this Section is to maximize the received SNR in order to improve the system error-rate performance.
  • Downloaded on November 19, 2008 at 19:08 from IEEE Xplore.
  • Assume Note that their optimization problem has a similar form to that in [17].
  • This shows that signal reception should be in the eigen-modes of the channel covariance matrix and effectively transforms the SIMO channel configuration into a set of Bk parallel and independent subchannels with the bth subchannel having a gain of |hk(b)|2δbµb,k.
  • Without going to the extend of rewriting all (16)-(18) for this case, the authors can first re-arrange Nrσ 2 k εs|hk(j)|2δj ’s in a non-descending order.

IV. NUMERICAL RESULTS

  • In this Section the authors provide bit-error-rate (BER) and symbolerror-rate (SER) curves for the proposed transmission schemes in broadband frequency-selective channels.
  • In their simulation, the following parameters and assumptions were adopted: the spatial channel correlation is modelled using the space-time channel with hyperbolically distributed scatterers in [18], Nc = 512, Nt = 1 (except for ST-OFDM, where two transmit antennas were used ), and QPSK baseband modulation is employed for both Figs. 5 and 6. In Fig. 5, the authors plot simulation results for systems with two receive antennas and different diversity coding schemes, i.e., OFDM with Alamouti’s space- Authorized licensed use limited to: RMIT University.
  • Downloaded on November 19, 2008 at 19:08 from IEEE Xplore.
  • Comparing the curves that correspond to these three schemes, it is clear that both ST-OFDM and FT-OFDM systems give significant error-rate improvement over the uncodedOFDM.
  • It is observed that better performance curves can be obtained in channels with higher spatial correlations for FT-Beam structure.

V. CONCLUSIONS

  • An adaptive transceiver structure that combines a new diversity coding scheme and receiver beamforming for uplink SIMO-OFDM transmission is investigated.
  • By utilizing the concept of subchannel grouping, FT coding that provides frequency diversity in the broadband wireless channel has proven to be an effective means of signal transmission for MS with a single antenna.
  • It is shown that adaptive eigenbeamforming at the receiver (to handle uplink signals in the eigen-modes of the correlation matrix) maximizes the received SNR and improves the system error-rate performance of the FT-coded OFDM system.

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Adaptive receiver beamforming for diversity
coded OFDM systems: Maximum SNR design
Lin, Kevin; Hussain, Zahir
https://researchrepository.rmit.edu.au/discovery/delivery/61RMIT_INST:ResearchRepository/12246722010001341?l#13248427710001341
Lin, & Hussain, Z. (2005). Adaptive receiver beamforming for diversity coded OFDM systems: Maximum
SNR design. Proceedings of the Tencon 2005 IEEE Region 10 Conference.
https://doi.org/10.1109/TENCON.2005.301115
Published Version: https://doi.org/10.1109/TENCON.2005.301115
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Adaptive Receiver Beamforming for Diversity
Coded OFDM Systems: Maximum SNR Design
Kevin H. Lin and Zahir M. Hussain
School of Electrical and Computer Engineering, RMIT University, Melbourne, Australia
Emails: s9510490@student.rmit.edu.au; zmhussain@ieee.org
Abstract Over the recent years, advance multiantenna trans-
mission schemes have attracted considerable interest due to
their potential benefits in improving the system capacity and
error-rate performance. As a result, space-time coding and
transmit beamforming have emerged as the two most promising
techniques. Because of limited space at the mobile station (MS)
and the fact that download intensive services are to be introduced
in the next generation of cellular systems, most of research
efforts have been pouring on transmit diversity techniques. In this
paper, we focus on adaptive uplink transmission and reception
techniques for wireless communications and introduce a new
frequency-time encoding scheme that can be used to exploit
frequency diversity branches for broadband OFDM systems with
only one antenna at the MS. By incorporating this with receive
beamforming at the base station (BS), the instantaneous signal-
to-noise ratio (SNR) is maximized and the system error-rate
performance is then further improved. Numerical results showed
that systems employed the proposed transceiver structure have
a 4-dB improvement over the conventional space-time coding
scheme when two receive antennas are used.
I. INTRODUCTION
Signal transmission in multi-input multi-output (MIMO)
systems that employs more than one antennas at the transmitter
and the receiver has shown to be effective in exploiting
spatial diversified paths of wireless channels [1]-[2] and in-
creasing both system capacity and error-rate performance. In
particular, space-time coding includes both space-time block
coding [3]-[4] and trellis coding [5] had gained a significant
attention due to their superior performance and simplicity
of transceiver design over other known techniques. However,
their performance improvements are based on the assumption
that the arriving multipath signals are sufficiently uncorrelated.
In cellular communications, due to close spacing between
antenna elements at the base station (BS), signal paths are
often correlated to some degree. As a consequence, coherent
deep fade between propagation signal paths is unavoidable
and studies have shown that signal correlation can degrade
the system performance significantly [6], [7].
The application of space-time coding to orthogonal fre-
quency division multiplexing (OFDM) systems was first in-
troduced in [8]. Motivated by the presence of additional mul-
tipath diversity offered by frequency-selectivity in broadband
wireless channels, space-frequency (SF) [7] and space-time-
frequency (STF) coding [9] were introduced. Extending from
their work, the combination of diversity coding schemes with
transmit beamforming was investigated for broadband OFDM
systems [10]-[11]. However, all of these transmission schemes
are for downlink application with multiple transmit antennas.
In this paper we propose a new frequency-time (FT) encod-
ing scheme and combine it with receiver beamforming (denote
by FT-Beam) to maximize the received signal-to-noise (SNR)
for OFDM systems with single transmit antenna. By knowing
subchannel gains at the mobile station (MS), we utilize the
concept of subchannel grouping in [12] and perform FT en-
coding of existing space-time codes across OFDM subcarriers
to achieve transmit diversity in the frequency domain. With the
effective use of beamforming at the multiantennas base station
(BS) receiver, the optimal adaptive beam-mapping weights is
applied to maximize the instantaneous SNR, and thus, system
error-rate performance during uplink transmission in a single-
input multi-output (SIMO) channel is further enhanced.
Notation used: (·)
, (·)
T
, and (·)
H
are complex conjugate,
vector transposition, and Hermitian transposition, respectively.
·
F
is the Frobenius norm;
A stands for Hermitian square
root of matrix A; det(·) denotes the determinant; E{·} is
the expectation operator. Finally, capital (small) bold letters
represent matrices (vectors).
II. S
YSTEM MODEL
Consider an uplink cellular communication scenario em-
ploying an N
c
frequency tone OFDM system with a single
transmit antenna at the MS and N
r
receive antennas at the
BS over a frequency-selective fading channel. It is assumed
that the channel coherent bandwidth is larger than the band-
width of each subcharrier; we thus consider the corresponding
subchannel to be frequency-flat. In Fig. 1, we depicted a
general structure of this OFDM system and combined with
the proposed adaptive transceiver structure. In this work, it
is also assumed that the system operates in a typical cellular
environment where the BS antennas are placed at the building
roof-top in an unobstructed manner. It is stated in [13] that
signal transmission in such an environment over a multipath
channel will lead to partially correlated signal paths in the
spatial domain arriving at the BS. Next, assume that a uniform
linear array (ULA) configuration is used for N
r
BS antennas
with a spacing of d meters. The normalized correlation matrix
that specifies the correlation between antenna elements is
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Input
Data
Symbols
Adaptive
Processor
Recovered
Data
Symbols
1
N
r
IFFT
FFT
FFT
Frequency
-Time
Encoder
S
/
P
Antenna
Weight
Mapper
P
/
S
Frequency
-Time
Decoder
Fig. 1. General structure of FT-Beam OFDM system with adaptive
receive-beamforming. Bold arrows represent multi-line signals.
defined in [14] as
R
r
=
1
L
L
=1
a(θ
)a
H
(θ
) (1)
where L denotes the number of dominant resolvable paths
and a(θ
):=[1,e
,e
j2β
, ··· ,e
j(N
r
1)β
]
T
is the array prop-
agation vector for the th tap with an angle-of-arrival (AoA)
of θ
. β =[2π · d · sin(θ
)], λ being the carrier frequency
wavelength. In general, R
r
is a nonnegative-definite Hermitian
matrix and the eigenvalue-decomposition (EVD) of R
r
can
be expressed as VR
r
V
H
= , where V =[υ
1
, ··· , υ
N
t
] is
a unitary matrix with columns that are the eigenvectors and
= diag[δ
1
, ···
n
, ···
N
r
] is a diagonal matrix contains
the corresponding eigenvalues. Without loss of generality, we
assume that δ
n
s are ordered in a non-increasing fashion:
δ
1
δ
2
···δ
N
r
0.
Let us denote the correlated SIMO channel frequency re-
sponse vector for the k
th
subcarrier as h
k
C
1×N
r
.Thej
th
element, which represents the subchannel gain between the
transmit and the j
th
receive antenna, is defined as h
k
(j):=
g
j
f
k
, where g
j
=[g
j
(1), ··· ,g
j
(L 1)] is the channel im-
pulse response vector with independent circularly symmetric
complex Gaussian random variables from CN(0
2
h
) and
f
k
=[1,e
j2π(k1)/N
c
, ··· ,e
j2π(k1)τ
L1
/N
c
]
T
is the cor-
responding discrete Fourier transform coefficients. According
to [13], the channel frequency response vector can also be
expressed as h
k
= h
k
R
r
, where h
k
can be thought as a pre-
whitened channel vector. Furthermore, quasi-static fading is
also assumed throughout the duration of one FTBC codeword
length but fading may vary from one block to another.
A. Subchannel Grouping & Frequency-Time Encoding
The concept of subchannel grouping, sometimes referred
as subcarrier grouping, was originally used in adaptive mod-
ulation scheme in [10] to reduce processing complexity by
grouping subcarriers or subchannels that are within one chan-
nel coherent bandwidth and having a similar fading gain.
group 1 group 2 group 3 group N
g
group i
f
1
f
2
f
3
f
4
f
5
f
6
f
Nc
f
k
;;
;;
;
Fig. 2. An illustration of subchannel grouping for OFDM systems
with N
c
frequency tones and the channel coherent bandwidth is
equivalent to the bandwidth of 3 subcarriers.
In [9] and [12], subcarrier grouping is used for grouping
frequencies that are approximately one coherent bandwidth
apart to perform STF coding in OFDM systems to exploit
both spatial and multipath diversity. In this work, we utilize the
concept of subchannel grouping in [12] by treating subcarriers
that are having different fading gains as additional antennas at
the MS. By doing so, we can then directly apply the space-
time codes in [3]-[5], [15] in our system by spreading symbol
energy across OFDM frequencies instead of antennas. An
illustratation of this subchannel grouping concept is shown
in Fig. 2, where the channel coherent bandwidth is assumed
to be equivalent to three frequency tones and subchannels that
are having different fading gains are grouped together. Thus,
the number of subchannels (subcarriers) that are in one group
depends on the spatial dimension of the original space-time
code.
Let us denote N
g
as the total number of groups as a
result of this sub-channel grouping process. If the well-known
Alamouti’s space-time block code in [3]
Space
−−−−−−−
Time
s
1
s
2
s
2
s
1
,
(which has a spatial dimension of 2) is used for the FT
encoding process, then N
g
= N
c
/2. An example of this FT
encoding output is shown in Fig. 3, where baseband modulated
data symbols s
1
and s
2
are FT-encoded across two subcarriers
f
1
and f
4
(c.f. Fig. 2) as well as two OFDM symbol periods
n =0and n =1(n being the time index in mod-2 sense).
Similarly, data symbols s
3
and s
4
are encoded in group 2,
while s
5
and s
6
are encoded in group 3. Note that other STBC
matrices in [4] and STTC technique in [5] can be applied in
the same way, but with different subchannel grouping sizes.
B. Receiver Beamforming
At the receiver, discrete Fourier transformation is applied to
the noisy samples of SIMO signals arriving at N
r
antennas.
Assume ideal symbol-time sampling and carrier synchroniza-
tion, the discrete time baseband equivalent form of system
input-output equation can be expressed as
r
k
= x
k
h
k
R
r
+ e
k
, (2)
where x
k
is an FT encoded data symbol transmitted on the k
th
subcarrier, and e
k
is a additive white Gaussian noise vector
with each element having zero mean and σ
2
k
variance.
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IFFT
Timing index, n
Frequency
n=0
f
1
f
2
f
3
f
4
-s
2
*
-s
4
*
-s
6
*
s
1
*
s
3
*
s
5
*
f
5
f
6
f
Nc
n=1
s
1
s
3
s
5
s
2
s
4
s
6
Fig. 3. Proposed FT encoding of Alamouti’s space-time block code
with only one transmit antenna.
Estimation of the channel fading gains is carried out at the
adaptive processor (AP) by correlating pilot tones embedded
in the transmitted signal. Results are then used for generating
the antenna weighting matrix to maximize the received SNR.
The antenna weight mapping process is performed across N
r
receive antennas. Mathematically, it can be written as
y
k
= r
k
W
k
= x
k
h
k
R
r
W
k
+ e
k
W
k
, (3)
where W
k
=
U
k
W
H
is the weighting matrix for the k
th
subcarrier, W =[w
1
, ··· , w
j
, ··· , w
N
r
]
T
C
N
r
×N
r
, and
w
j
is the steering weight vector for the j
th
receive antenna.
The matrix U
k
= diag[µ
2
1,k
, ···
2
N
r
,k
] contains the power
splitting ratio for the steering weight vectors.
C. Frequency-Time Decoding
To recover original data, channel estimation results are also
used for the FT decoding process. Since frequency subcarriers
are treated as additional transmit antennas at the transmitter FT
encoding, the decoding algorithm is divided into a combining
stage and a maximum likelihood decoding (MLD) stage.
The combining stage is simply adding the received signals
in different subcarriers that are within the same subchannel
group.
Denote i(m) as the frequency index for subchannel group
i. Using the previous example (c.r. Fig. 2 & 3), the frequency
index of subchannel group 1 will read 1(1), which corresponds
to subcarrier 1, and 1(2) corresponds to subcarrier 4. Thus,
without the presumption that Alamouti’s space-time block
code is used for the FT encoding, the combining process can
be expressed as
y
i
(n)=
N
c
/N
g
m=1
y
i(m)
(n). (4)
Following the maximum likelihood detection rule in [15], FT
decoding of symbol s
1
of Alamouti’s code amounts minimiz-
ing the decision matrix in subchannel group i as
N
r
j=1
y
i,j
(n)h
i(1)
(j)+y
i,j
(n +1)h
i(2)
(j)
s
1
2
+
1+
N
r
j=1
2
m=1
|h
i(m)
(j)|
2
|s
1
|
2
, (5)
where h
i(m)
(j) and y
i,j
(n) denotes the j
th
entry of
h
i(m)
R
r
W
i(m)
and y
i
(n), respectively. The decision matrix
N
r
j=1
y
i,j
(n)h
i(2)
(j) y
i,j
(n +1)h
i(1)
(j)
s
2
2
+
1+
N
r
j=1
2
m=1
|h
i(m)
(j)|
2
|s
2
|
2
, (6)
is used for decoding s
2
. Similarly, the above MLD expressions
can be easily extended and used for decoding of other space-
time block codes in [4].
D. Signal-to-Noise Ratio (SNR)
Given that the average transmitted energy during one
OFDM-symbol interval is
E{x
k
} = E{|s
n
|
2
} = ε
s
, (7)
the received SNR at the k
th
subcarrier for the detection of x
k
has the form:
γ
k
=
ε
s
h
k
R
r
W
k
2
F
σ
2
k
W
k
2
F
. (8)
Denote |s
n
˜s
n
| as the minimum distance between the
underlying constellation symbols, then the symbol energy for
both QAM and PSK modulation schemes are given in [16] as
ε
s
=
(M 1)|s
n
˜s
n
|
2
QAM
6
, for QAM (9)
ε
s
=
|s
n
˜s
n
|
2
PSK
4sin
2
(π/M)
, for PSK. (10)
III. O
PTIMAL ANTENNA WEIGHTING MAT RIX:MAXIMUM
SNR DESIGN
The objective of this Section is to maximize the received
SNR in order to improve the system error-rate performance.
Effectively, this amounts choosing the weight mapping matrix
that maximize (8) by solving the following cost function:
max
W
k
J = det
I
N
r
+
ε
s
h
k
R
r
W
k
2
F
σ
2
k
W
k
2
F
= det
I
N
r
+
ε
s
h
k
R
r
W
k
W
H
k
R
r
H
h
H
k
σ
2
k
W
k
2
F
subject to : W
k
2
F
= N
r
. (11)
Equivalently, this can be re-written as
max
W
k
J = det
I
N
r
+
ε
s
h
k
ΛΦ
Λ
H
h
H
k
N
r
σ
2
k
, (12)
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where Φ
k
= V
H
U
k
W
H
W
U
k
H
V. By using Hadamard
inequality, the optimization problem (12) can be rewritten as
J det
I
N
r
+
ε
s
h
k
ΛΦ
k
Λ
H
h
H
k
N
r
σ
2
k
(13)
and the equality is achieved if and only if Φ
k
is a diagonal
matrix. Assume Φ
k
= diag[φ
1,k
2,k
, ···
N
r
,k
], hence (13)
becomes
J =
N
r
j=1
1+
ε
s
N
r
σ
2
k
|h
k
(j)|
2
δ
j
φ
j,k
, (14)
where
h
k
(j) is the j
th
element of h
k
. Note that our optimiza-
tion problem has a similar form to that in [17]. Although the
water-filling strategy was originally used for enhancing the
channel capacity, in this work we utilize it for maximizing
the instantaneous received SNR. Following [17], we arrive
at an initial solution for φ
j,k
’s as φ
j,k
= ξ
N
r
σ
2
k
ε
s
|h
k
(j)|
2
δ
j
.
However, depending on the channel fading gains and the
receiver noise variance, this solution may not satisfy the
condition φ
j,k
0, j due to the constraint
N
r
j=1
φ
j,k
= N
r
.
Thus we introduce a special notation (x)
+
denoting max(x, 0).
Now φ
j,k
will be
φ
j,k
=
ξ
N
r
σ
2
k
ε
s
|h
k
(j)|
2
δ
j
+
. (15)
Recall that δ
j
s are arranged in a non-increasing or-
der, for now, we assume φ
j,k
s are also arranged in
the same order, φ
1,k
φ
2,k
···φ
N
t
,k
, as long as
|
h
k
(1)|
2
≥|h
k
(2)|
2
··· |h
k
(N
r
)|
2
.LetB
k
represent
the number of non-zero φ
j,k
s, then Φ
k
becomes Φ
k
=
diag [φ
1,k
, ···
B
k
,k
, 0
B
k
+1,k
, ··· , 0
N
r
,k
], where B
k
N
r
.
Next, based on the power splitting constraint, we know that ξ
is chosen so that
B
k
b=1
φ
b,k
=
B
k
b=1
ξ
N
r
σ
2
k
ε
s
|h
k
(b)|
2
δ
b
= N
r
. (16)
Inverting (16), expressing it as a function of ξ, and substi-
tuting into (15) we get
φ
j,k
=
1
B
k
+
1
B
k
B
k
b=1
N
r
σ
2
k
ε
s
|h
k
(b)|
2
δ
b
N
r
σ
2
k
ε
s
|h
k
(j)|
2
δ
j
+
. (17)
Up to this point, we still need to find a value for B
k
.To
find the optimal value for B
k
,weseth
k
(j) and δ
j
to h
k
(B
k
)
and δ
B
k
, respectively, then test the following inequality
1
B
k
+
1
B
k
B
k
b=1
N
r
σ
2
k
ε
s
|h
k
(b)|
2
δ
b
N
r
σ
2
k
ε
s
|h
k
(B
k
)|
2
δ
B
k
> 1 (18)
for B
k
=1, ··· ,N
r
. Thus, the optimum value for B
k
is the largest value that satisfies the inequality, and signal
transmission utilizing a number of beams that is greater than
B
k
will incur a loss in the potential performance gain.
Now the optimization of (13) with respect to Φ
k
for a
given h
k
and subject to a power constraint can be written as
Fig. 4. Water-filling for re-ordered N
r
sub-channels.
J
B
k
b=1
1+
ε
s
N
r
σ
2
k
|h
k
(b)|
2
δ
b
φ
b,k
with equality achieved
if and only if W
k
is chosen as W
k
=
k
V
H
, where
W = V and U
k
= Φ
k
. Hence, µ
j,k
=
φ
j,k
.This
shows that signal reception should be in the eigen-modes of
the channel covariance matrix and effectively transforms the
SIMO channel configuration into a set of B
k
parallel and
independent subchannels with the b
th
subchannel having a
gain of |
h
k
(b)|
2
δ
b
µ
b,k
. In case when the previous assumption
|
h
k
(1)|
2
≥|h
k
(2)|
2
··· |h
k
(N
r
)|
2
does not hold, then
N
r
σ
2
k
ε
s
|h
k
(1)|
2
δ
1
N
r
σ
2
k
ε
s
|h
k
(2)|
2
δ
2
···
N
r
σ
2
k
ε
s
|h
k
(N
r
)|
2
δ
N
r
will not
be valid. Hence, (16)-(18) will no longer be applicable since
b is indexing |
h
k
(j)|
2
s in a non-descending order. Without
going to the extend of rewriting all (16)-(18) for this case, we
can first re-arrange
N
r
σ
2
k
ε
s
|h
k
(j)|
2
δ
j
s in a non-descending order.
This guarantees that φ
1
φ
2
··· φ
N
r
.Useb to index
N
r
σ
2
k
ε
s
|h
k
(j)|
2
δ
j
s that are less than ξ (as shown in Fig. 4) instead of
δ
j
s, such that φ
j,k
s and the optimum B
k
can still be found
by using (17) and (18), respectively. Then signal reception
is now in the directions of B
k
eigen-beams (not necessary
corresponding to the first B
k
eigenvalues) that give highest
instantaneous received SNR gain.
IV. N
UMERICAL RESULTS
In this Section we provide bit-error-rate (BER) and symbol-
error-rate (SER) curves for the proposed transmission schemes
in broadband frequency-selective channels. In our simulation,
the following parameters and assumptions were adopted: the
spatial channel correlation is modelled using the space-time
channel with hyperbolically distributed scatterers in [18],
N
c
= 512, N
t
=1(except for ST-OFDM, where two transmit
antennas were used ), and QPSK baseband modulation is
employed for both Figs. 5 and 6. In Fig. 5, we plot simulation
results for systems with two receive antennas and different
diversity coding schemes, i.e., OFDM with Alamouti’s space-
Authorized licensed use limited to: RMIT University. Downloaded on November 19, 2008 at 19:08 from IEEE Xplore. Restrictions apply.

References
More filters

Book
Thomas M. Cover1, Joy A. Thomas2Institutions (2)
01 Jan 1991-
TL;DR: The author examines the role of entropy, inequality, and randomness in the design of codes and the construction of codes in the rapidly changing environment.
Abstract: Preface to the Second Edition. Preface to the First Edition. Acknowledgments for the Second Edition. Acknowledgments for the First Edition. 1. Introduction and Preview. 1.1 Preview of the Book. 2. Entropy, Relative Entropy, and Mutual Information. 2.1 Entropy. 2.2 Joint Entropy and Conditional Entropy. 2.3 Relative Entropy and Mutual Information. 2.4 Relationship Between Entropy and Mutual Information. 2.5 Chain Rules for Entropy, Relative Entropy, and Mutual Information. 2.6 Jensen's Inequality and Its Consequences. 2.7 Log Sum Inequality and Its Applications. 2.8 Data-Processing Inequality. 2.9 Sufficient Statistics. 2.10 Fano's Inequality. Summary. Problems. Historical Notes. 3. Asymptotic Equipartition Property. 3.1 Asymptotic Equipartition Property Theorem. 3.2 Consequences of the AEP: Data Compression. 3.3 High-Probability Sets and the Typical Set. Summary. Problems. Historical Notes. 4. Entropy Rates of a Stochastic Process. 4.1 Markov Chains. 4.2 Entropy Rate. 4.3 Example: Entropy Rate of a Random Walk on a Weighted Graph. 4.4 Second Law of Thermodynamics. 4.5 Functions of Markov Chains. Summary. Problems. Historical Notes. 5. Data Compression. 5.1 Examples of Codes. 5.2 Kraft Inequality. 5.3 Optimal Codes. 5.4 Bounds on the Optimal Code Length. 5.5 Kraft Inequality for Uniquely Decodable Codes. 5.6 Huffman Codes. 5.7 Some Comments on Huffman Codes. 5.8 Optimality of Huffman Codes. 5.9 Shannon-Fano-Elias Coding. 5.10 Competitive Optimality of the Shannon Code. 5.11 Generation of Discrete Distributions from Fair Coins. Summary. Problems. Historical Notes. 6. Gambling and Data Compression. 6.1 The Horse Race. 6.2 Gambling and Side Information. 6.3 Dependent Horse Races and Entropy Rate. 6.4 The Entropy of English. 6.5 Data Compression and Gambling. 6.6 Gambling Estimate of the Entropy of English. Summary. Problems. Historical Notes. 7. Channel Capacity. 7.1 Examples of Channel Capacity. 7.2 Symmetric Channels. 7.3 Properties of Channel Capacity. 7.4 Preview of the Channel Coding Theorem. 7.5 Definitions. 7.6 Jointly Typical Sequences. 7.7 Channel Coding Theorem. 7.8 Zero-Error Codes. 7.9 Fano's Inequality and the Converse to the Coding Theorem. 7.10 Equality in the Converse to the Channel Coding Theorem. 7.11 Hamming Codes. 7.12 Feedback Capacity. 7.13 Source-Channel Separation Theorem. Summary. Problems. Historical Notes. 8. Differential Entropy. 8.1 Definitions. 8.2 AEP for Continuous Random Variables. 8.3 Relation of Differential Entropy to Discrete Entropy. 8.4 Joint and Conditional Differential Entropy. 8.5 Relative Entropy and Mutual Information. 8.6 Properties of Differential Entropy, Relative Entropy, and Mutual Information. Summary. Problems. Historical Notes. 9. Gaussian Channel. 9.1 Gaussian Channel: Definitions. 9.2 Converse to the Coding Theorem for Gaussian Channels. 9.3 Bandlimited Channels. 9.4 Parallel Gaussian Channels. 9.5 Channels with Colored Gaussian Noise. 9.6 Gaussian Channels with Feedback. Summary. Problems. Historical Notes. 10. Rate Distortion Theory. 10.1 Quantization. 10.2 Definitions. 10.3 Calculation of the Rate Distortion Function. 10.4 Converse to the Rate Distortion Theorem. 10.5 Achievability of the Rate Distortion Function. 10.6 Strongly Typical Sequences and Rate Distortion. 10.7 Characterization of the Rate Distortion Function. 10.8 Computation of Channel Capacity and the Rate Distortion Function. Summary. Problems. Historical Notes. 11. Information Theory and Statistics. 11.1 Method of Types. 11.2 Law of Large Numbers. 11.3 Universal Source Coding. 11.4 Large Deviation Theory. 11.5 Examples of Sanov's Theorem. 11.6 Conditional Limit Theorem. 11.7 Hypothesis Testing. 11.8 Chernoff-Stein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the Cram-er-Rao Inequality. Summary. Problems. Historical Notes. 12. Maximum Entropy. 12.1 Maximum Entropy Distributions. 12.2 Examples. 12.3 Anomalous Maximum Entropy Problem. 12.4 Spectrum Estimation. 12.5 Entropy Rates of a Gaussian Process. 12.6 Burg's Maximum Entropy Theorem. Summary. Problems. Historical Notes. 13. Universal Source Coding. 13.1 Universal Codes and Channel Capacity. 13.2 Universal Coding for Binary Sequences. 13.3 Arithmetic Coding. 13.4 Lempel-Ziv Coding. 13.5 Optimality of Lempel-Ziv Algorithms. Compression. Summary. Problems. Historical Notes. 14. Kolmogorov Complexity. 14.1 Models of Computation. 14.2 Kolmogorov Complexity: Definitions and Examples. 14.3 Kolmogorov Complexity and Entropy. 14.4 Kolmogorov Complexity of Integers. 14.5 Algorithmically Random and Incompressible Sequences. 14.6 Universal Probability. 14.7 Kolmogorov complexity. 14.9 Universal Gambling. 14.10 Occam's Razor. 14.11 Kolmogorov Complexity and Universal Probability. 14.12 Kolmogorov Sufficient Statistic. 14.13 Minimum Description Length Principle. Summary. Problems. Historical Notes. 15. Network Information Theory. 15.1 Gaussian Multiple-User Channels. 15.2 Jointly Typical Sequences. 15.3 Multiple-Access Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between Slepian-Wolf Encoding and Multiple-Access Channels. 15.6 Broadcast Channel. 15.7 Relay Channel. 15.8 Source Coding with Side Information. 15.9 Rate Distortion with Side Information. 15.10 General Multiterminal Networks. Summary. Problems. Historical Notes. 16. Information Theory and Portfolio Theory. 16.1 The Stock Market: Some Definitions. 16.2 Kuhn-Tucker Characterization of the Log-Optimal Portfolio. 16.3 Asymptotic Optimality of the Log-Optimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the Log-Optimal Portfolio. 16.7 Universal Portfolios. 16.8 Shannon-McMillan-Breiman Theorem (General AEP). Summary. Problems. Historical Notes. 17. Inequalities in Information Theory. 17.1 Basic Inequalities of Information Theory. 17.2 Differential Entropy. 17.3 Bounds on Entropy and Relative Entropy. 17.4 Inequalities for Types. 17.5 Combinatorial Bounds on Entropy. 17.6 Entropy Rates of Subsets. 17.7 Entropy and Fisher Information. 17.8 Entropy Power Inequality and Brunn-Minkowski Inequality. 17.9 Inequalities for Determinants. 17.10 Inequalities for Ratios of Determinants. Summary. Problems. Historical Notes. Bibliography. List of Symbols. Index.

42,928 citations


"Adaptive Receiver Beamforming for D..." refers result in this paper

  • ...Note that our optimization problem has a similar form to that in [17]....

    [...]


Journal ArticleDOI
Siavash Alamouti1Institutions (1)
TL;DR: This paper presents a simple two-branch transmit diversity scheme that provides the same diversity order as maximal-ratio receiver combining (MRRC) with one transmit antenna, and two receive antennas.
Abstract: This paper presents a simple two-branch transmit diversity scheme. Using two transmit antennas and one receive antenna the scheme provides the same diversity order as maximal-ratio receiver combining (MRRC) with one transmit antenna, and two receive antennas. It is also shown that the scheme may easily be generalized to two transmit antennas and M receive antennas to provide a diversity order of 2M. The new scheme does not require any bandwidth expansion or any feedback from the receiver to the transmitter and its computation complexity is similar to MRRC.

13,447 citations


Journal ArticleDOI
Emre Telatar1Institutions (1)
01 Nov 1999-
Abstract: We investigate the use of multiple transmitting and/or receiving antennas for single user communications over the additive Gaussian channel with and without fading. We derive formulas for the capacities and error exponents of such channels, and describe computational procedures to evaluate such formulas. We show that the potential gains of such multi-antenna systems over single-antenna systems is rather large under independenceassumptions for the fades and noises at different receiving antennas.

12,396 citations


Journal ArticleDOI
G. J. Foschini1, M. J. Gans1Institutions (1)
Abstract: This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bit-rates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multi-element array (MEA) technology, that is processing the spatial dimension (not just the time dimension) to improve wireless capacities in certain applications. Specifically, we present some basic information theory results that promise great advantages of using MEAs in wireless LANs and building to building wireless communication links. We explore the important case when the channel characteristic is not available at the transmitter but the receiver knows (tracks) the characteristic which is subject to Rayleigh fading. Fixing the overall transmitted power, we express the capacity offered by MEA technology and we see how the capacity scales with increasing SNR for a large but practical number, n, of antenna elements at both transmitter and receiver. We investigate the case of independent Rayleigh faded paths between antenna elements and find that with high probability extraordinary capacity is available. Compared to the baseline n = 1 case, which by Shannon‘s classical formula scales as one more bit/cycle for every 3 dB of signal-to-noise ratio (SNR) increase, remarkably with MEAs, the scaling is almost like n more bits/cycle for each 3 dB increase in SNR. To illustrate how great this capacity is, even for small n, take the cases n = 2, 4 and 16 at an average received SNR of 21 dB. For over 99% of the channels the capacity is about 7, 19 and 88 bits/cycle respectively, while if n = 1 there is only about 1.2 bit/cycle at the 99% level. For say a symbol rate equal to the channel bandwith, since it is the bits/symbol/dimension that is relevant for signal constellations, these higher capacities are not unreasonable. The 19 bits/cycle for n = 4 amounts to 4.75 bits/symbol/dimension while 88 bits/cycle for n = 16 amounts to 5.5 bits/symbol/dimension. Standard approaches such as selection and optimum combining are seen to be deficient when compared to what will ultimately be possible. New codecs need to be invented to realize a hefty portion of the great capacity promised.

10,358 citations


Journal ArticleDOI
TL;DR: A generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas and it is shown that many of the codes presented here are optimal in this sense.
Abstract: We introduce space-time block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a space-time block code and the encoded data is split into n streams which are simultaneously transmitted using n transmit antennas. The received signal at each receive antenna is a linear superposition of the n transmitted signals perturbed by noise. Maximum-likelihood decoding is achieved in a simple way through decoupling of the signals transmitted from different antennas rather than joint detection. This uses the orthogonal structure of the space-time block code and gives a maximum-likelihood decoding algorithm which is based only on linear processing at the receiver. Space-time block codes are designed to achieve the maximum diversity order for a given number of transmit and receive antennas subject to the constraint of having a simple decoding algorithm. The classical mathematical framework of orthogonal designs is applied to construct space-time block codes. It is shown that space-time block codes constructed in this way only exist for few sporadic values of n. Subsequently, a generalization of orthogonal designs is shown to provide space-time block codes for both real and complex constellations for any number of transmit antennas. These codes achieve the maximum possible transmission rate for any number of transmit antennas using any arbitrary real constellation such as PAM. For an arbitrary complex constellation such as PSK and QAM, space-time block codes are designed that achieve 1/2 of the maximum possible transmission rate for any number of transmit antennas. For the specific cases of two, three, and four transmit antennas, space-time block codes are designed that achieve, respectively, all, 3/4, and 3/4 of maximum possible transmission rate using arbitrary complex constellations. The best tradeoff between the decoding delay and the number of transmit antennas is also computed and it is shown that many of the codes presented here are optimal in this sense as well.

7,263 citations


"Adaptive Receiver Beamforming for D..." refers background or methods in this paper

  • ...Note that other STBC matrices in [4] and STTC technique in [5] can be applied in the same way, but with different subchannel grouping sizes....

    [...]

  • ...In particular, space-time coding includes both space-time block coding [3]-[4] and trellis coding [5] had gained a significant attention due to their superior performance and simplicity of transceiver design over other known techniques....

    [...]

  • ...Similarly, the above MLD expressions can be easily extended and used for decoding of other spacetime block codes in [4]....

    [...]


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