Please do not remove this page
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
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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
jβ
,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, 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
, ··· ,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 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,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, 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
, ··· ,µ
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
,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
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
, ··· ,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. 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
= 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 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-
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