Adaptive Receiver Beamforming for Diversity Coded OFDM Systems: Maximum SNR Design
01 Nov 2005pp 15
TL;DR: A new frequencytime 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 4dB improvement over the conventional spacetime 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 errorrate performance. As a result, spacetime 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 frequencytime 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 tonoise ratio (SNR) is maximized and the system errorrate performance is then further improved. Numerical results showed that systems employed the proposed transceiver structure have a 4dB improvement over the conventional spacetime 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)
Jump to: [Introduction] – [II. SYSTEM MODEL] – [A. Subchannel Grouping & FrequencyTime Encoding] – [B. Receiver Beamforming] – [C. FrequencyTime Decoding] – [D. SignaltoNoise Ratio (SNR)] – [III. OPTIMAL ANTENNA WEIGHTING MATRIX: MAXIMUM SNR DESIGN] – [IV. NUMERICAL RESULTS] and [V. CONCLUSIONS]
Introduction
 I. INTRODUCTION Signal transmission in multiinput multioutput (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 errorrate performance.
 Their performance improvements are based on the assumption that the arriving multipath signals are sufficiently uncorrelated.
 The application of spacetime 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 beammapping weights is applied to maximize the instantaneous SNR, and thus, system errorrate performance during uplink transmission in a singleinput multioutput (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 frequencyflat.
 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, quasistatic fading is also assumed throughout the duration of one FTBC codeword length but fading may vary from one block to another.
A. Subchannel Grouping & FrequencyTime 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 wellknown Alamouti’s spacetime 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 symboltime sampling and carrier synchronization, the discrete time baseband equivalent form of system inputoutput 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. FrequencyTime 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 spacetime 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. SignaltoNoise Ratio (SNR)
 Given that the average transmitted energy during one OFDMsymbol interval is E{xk} = E{sn2} = 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 − s̃n2QAM 6 , for QAM (9) εs = sn − s̃n2PSK 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 errorrate 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 eigenmodes 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 rearrange Nrσ 2 k εshk(j)2δj ’s in a nondescending order.
IV. NUMERICAL RESULTS
 In this Section the authors provide biterrorrate (BER) and symbolerrorrate (SER) curves for the proposed transmission schemes in broadband frequencyselective channels.
 In their simulation, the following parameters and assumptions were adopted: the spatial channel correlation is modelled using the spacetime channel with hyperbolically distributed scatterers in [18], Nc = 512, Nt = 1 (except for STOFDM, 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 STOFDM and FTOFDM systems give significant errorrate improvement over the uncodedOFDM.
 It is observed that better performance curves can be obtained in channels with higher spatial correlations for FTBeam structure.
V. CONCLUSIONS
 An adaptive transceiver structure that combines a new diversity coding scheme and receiver beamforming for uplink SIMOOFDM 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 eigenmodes of the correlation matrix) maximizes the received SNR and improves the system errorrate performance of the FTcoded 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
Downloaded On 2022/08/10 15:53:11 +1000
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Repository homepage: https://researchrepository.rmit.edu.au
Please do not remove this page
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
errorrate performance. As a result, spacetime 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
frequencytime 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
tonoise ratio (SNR) is maximized and the system errorrate
performance is then further improved. Numerical results showed
that systems employed the proposed transceiver structure have
a 4dB improvement over the conventional spacetime coding
scheme when two receive antennas are used.
I. INTRODUCTION
Signal transmission in multiinput multioutput (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 errorrate performance. In
particular, spacetime coding includes both spacetime 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 spacetime 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 frequencyselectivity in broadband
wireless channels, spacefrequency (SF) [7] and spacetime
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 frequencytime (FT) encod
ing scheme and combine it with receiver beamforming (denote
by FTBeam) to maximize the received signaltonoise (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 spacetime 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 beammapping weights is
applied to maximize the instantaneous SNR, and thus, system
errorrate performance during uplink transmission in a single
input multioutput (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 frequencyselective 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 frequencyﬂat. 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
rooftop 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 FTBeam OFDM system with adaptive
receivebeamforming. Bold arrows represent multiline 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
jβ
,e
j2β
, ··· ,e
j(N
r
−1)β
]
T
is the array prop
agation vector for the th tap with an angleofarrival (AoA)
of θ
. β =[2π · d · sin(θ
)]/λ, λ being the carrier frequency
wavelength. In general, R
r
is a nonnegativedeﬁnite Hermitian
matrix and the eigenvaluedecomposition (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
, ··· ,δ
n
, ··· ,δ
N
r
] is a diagonal matrix contains
the corresponding eigenvalues. Without loss of generality, we
assume that δ
n
’s are ordered in a nonincreasing 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,σ
2
h
) and
f
k
=[1,e
−j2π(k−1)/N
c
, ··· ,e
−j2π(k−1)τ
L−1
/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, quasistatic fading is
also assumed throughout the duration of one FTBC codeword
length but fading may vary from one block to another.
A. Subchannel Grouping & FrequencyTime 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 spacetime
code.
Let us denote N
g
as the total number of groups as a
result of this subchannel grouping process. If the wellknown
Alamouti’s spacetime 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 FTencoded 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 mod2 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 symboltime sampling and carrier synchroniza
tion, the discrete time baseband equivalent form of system
inputoutput 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 spacetime 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
, ··· ,µ
2
N
r
,k
] contains the power
splitting ratio for the steering weight vectors.
C. FrequencyTime 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 spacetime 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. SignaltoNoise Ratio (SNR)
Given that the average transmitted energy during one
OFDMsymbol 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 errorrate 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 rewritten 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
,φ
2,k
, ··· ,φ
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
waterﬁlling 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 nonincreasing 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 nonzero φ
j,k
’s, then Φ
k
becomes Φ
k
=
diag [φ
1,k
, ··· ,φ
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. Waterﬁlling for reordered N
r
subchannels.
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
= VΦ
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 eigenmodes 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 nondescending order. Without
going to the extend of rewriting all (16)(18) for this case, we
can ﬁrst rearrange
N
r
σ
2
k
ε
s
h
k
(j)
2
δ
j
’s in a nondescending 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
eigenbeams (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 biterrorrate (BER) and symbol
errorrate (SER) curves for the proposed transmission schemes
in broadband frequencyselective channels. In our simulation,
the following parameters and assumptions were adopted: the
spatial channel correlation is modelled using the spacetime
channel with hyperbolically distributed scatterers in [18],
N
c
= 512, N
t
=1(except for STOFDM, 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
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References
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•
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 DataProcessing 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 HighProbability 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 ShannonFanoElias 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 ZeroError 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 SourceChannel 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 ChernoffStein Lemma. 11.9 Chernoff Information. 11.10 Fisher Information and the CramerRao 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 LempelZiv Coding. 13.5 Optimality of LempelZiv 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 MultipleUser Channels. 15.2 Jointly Typical Sequences. 15.3 MultipleAccess Channel. 15.4 Encoding of Correlated Sources. 15.5 Duality Between SlepianWolf Encoding and MultipleAccess 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 KuhnTucker Characterization of the LogOptimal Portfolio. 16.3 Asymptotic Optimality of the LogOptimal Portfolio. 16.4 Side Information and the Growth Rate. 16.5 Investment in Stationary Markets. 16.6 Competitive Optimality of the LogOptimal Portfolio. 16.7 Universal Portfolios. 16.8 ShannonMcMillanBreiman 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 BrunnMinkowski 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]....
[...]
••
TL;DR: This paper presents a simple twobranch transmit diversity scheme that provides the same diversity order as maximalratio receiver combining (MRRC) with one transmit antenna, and two receive antennas.
Abstract: This paper presents a simple twobranch transmit diversity scheme. Using two transmit antennas and one receive antenna the scheme provides the same diversity order as maximalratio 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
••
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 multiantenna systems over singleantenna systems is rather large under independenceassumptions for the fades and noises at different receiving antennas.
12,396 citations
••
Abstract: This paper is motivated by the need for fundamental understanding of ultimate limits of bandwidth efficient delivery of higher bitrates in digital wireless communications and to also begin to look into how these limits might be approached. We examine exploitation of multielement 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 signaltonoise 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
••
TL;DR: A generalization of orthogonal designs is shown to provide spacetime 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 spacetime block coding, a new paradigm for communication over Rayleigh fading channels using multiple transmit antennas. Data is encoded using a spacetime 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. Maximumlikelihood 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 spacetime block code and gives a maximumlikelihood decoding algorithm which is based only on linear processing at the receiver. Spacetime 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 spacetime block codes. It is shown that spacetime block codes constructed in this way only exist for few sporadic values of n. Subsequently, a generalization of orthogonal designs is shown to provide spacetime 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, spacetime 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, spacetime 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, spacetime coding includes both spacetime 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]....
[...]