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Regular paper Transmitter diversity antenna
selection techniques for wireless channels
utilizing differential space-time block codes
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
Abstract— The paper deals with transmitter diversity antenna
selection techniques (ASTs) for wireless channels utilizing dif-
ferential space-time block codes (DSTBCs). The proposed
ASTs tend to maximize the signal-to-noise ratio (SNR) of those
channels. Particularly, we propose here the so-called gen-
eral (M
M
M,N
N
N;K
K
K)AST/DSTBC scheme for such channels. Then,
based on this AST, we propose two modified ASTs which are
more amenable to practical implementation, namely the re-
stricted (M
M
M,N
N
N;K
K
K)AST/DSTBC scheme and the (N
N
N+¯
N
N
N,N
N
N;K
K
K)
AST/DSTBC scheme. The restricted (M
M
M,N
N
N;K
K
K)AST/DSTBC
scheme provides relatively good bit error performance us-
ing only one feedback bit for transmission diversity purpose,
while the (N
N
N+¯
N
N
N,N
N
N;K
K
K)AST/DSTBC scheme shortens the time
required to process feedback information. These techniques
remarkably improve bit error rate (BER) performance of
wireless channels using DSTBCs with a limited number (typi-
cally 1 or 2) of training symbols per each coherent duration of
the channel. Simulations show that the proposed AST/DSTBC
schemes outperform the DSTBCs without antenna selection
even with only 1 training symbol.
Keywords—differential space-time modulation, differential
space-time block codes, diversity antenna selection, MIMO.
1. Introduction
The diversity combination of space-time codes (STCs) and
a closed loop antenna selection technique (AST) assisted
by a feedback channel to improve the performance of wire-
less channels in multiple input multiple output (MIMO)
systems has been intensively examined in literature for the
case of coherent detection, such as [5–10]. However, ASTs
for channels utilizing differential space-time block codes
(DSTBCs) with differential detection have not been con-
sidered yet. The backgrounds on DSTBCs can be found
in [11–17].
In this paper1, we propose some ASTs which tend to max-
imize the signal-to-noise ratio (SNR) for the channels us-
ing DSTBCs with arbitrary number Mof transmit antennas
(MTx antennas) and with Kreceive antennas (KRx anten-
nas). Particularly, we first propose an AST called the gen-
eral (M,N;K)AST/DSTBC where the transmitter selects
NTx antennas out of MTx antennas (M>N) to maximize
the channel SNR. The antenna selection (at the transmit-
ter) is based on the results of the comparison carried out
(at the receiver) between the instantaneous powers of sig-
1Related to the content in this paper are the published works [1–4].
nals which are received during the initial transmission. The
general (M,N;K)AST/DSTBC significantly improves the
performance of channels using DSTBCs. However, when
Mand Ngrow large, the number of feedback bits required
to inform the transmitter also grows large. This drawback
impedes the general (M,N;K)AST/DSTBC from practical
implementation if Mand Nare large.
The aforementioned drawback can be overcome by either
reducing the number of feedback bits or shortening the
time required to process feedback information. Based
on these observations, we modify the general (M,N;K)
AST/DSTBC and derive the two following ASTs which are
more amenable to practical implementation.
First, we propose the so-called restricted (M,N;K)
AST/DSTBC, which provides good bit error performance
using only 1 feedback bit for transmission diversity pur-
pose.
Then, we describe the so-called (N+¯
N,N;K)AST/DSTBC
which shortens the average time required to process feed-
back information in comparison with the general (M,N;K)
AST/DSTBC, where M=N+¯
N. This AST is first moti-
vated by the (N+1,N;K) AST/STBC which we mentioned
in [1] for channels using space-time block codes (STBCs)
with coherent detection. The background on STBCs can
be found in [18–21].
We show that DSTBCs associated with the proposed ASTs
provide much better bit error performance than that without
antenna selection. The proposed ASTs in this paper are the
generalization of our ASTs published in [2, 3]. The content
of this paper is also somewhat related to our published
papers [1, 4].
Although, the authors propose here the ASTs for a very
general case, where the system contains arbitrary numbers
of Tx and Rx antennas, it is important having in mind that it
is more practical to have diversity antennas installed at the
transmitter, e.g., a base station in mobile communication
systems, rather than at the hand-held, tiny receiver, such
as a mobile phone. It is well known that the installation
of more than 2Tx antennas in mobile phones is almost
impractical due to the battery life-time and the small size
of the phones.
Consequently, by using the term antenna selection in this
paper, we mean transmitter diversity antenna selection,
rather than receiver diversity antenna selection, i.e., all
KRx antennas are used without selection (although the
generalization of the proposed ASTs to receiver diversity
antenna selection is straightforward). It should be also
79
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
noted that the term differential space-time block codes
(DSTBCs) used throughout this paper means complex, or-
thogonal DSTBCs.
This paper is organized as follows.
Section 2 reviews the conventional DSTBCs mentioned in
literature and provides some remarks on the time-varying
Rayleigh fading channels where DSTBCs can be practically
used. In Section 3, we mention some notations and assump-
tions used throughout this paper. Section 4 starts with the
discussion on the criterion of antenna selection in channels
using STBCs and then analyzes our modifications to apply
to channels using DSTBCs. In Section 5, we propose the
general (M,N;K)AST/DSTBC. In Section 6.1, we propose
the restricted (M,N;K)AST/DSTBC. The (N+¯
N,N;K)
AST/DSTBC is proposed in Section 6.2. Section 7 provides
the mathematical expression of the relative time reduction
gained by the (N+¯
N,N;K) AST/DSTBC in comparison
with the general (M,N;K)AST/DSTBC. In Section 8, we
give some comments on the spatial diversity order of our
proposed ASTs. Simulation results are presented in Sec-
tion 9 and the paper is concluded by Section 10.
2. Reviews on DSTBCs
In this section, we review the conventional DSTBCs men-
tioned in literature and provide some remarks on the time-
varying Rayleigh fading channels where DSTBCs can be
practically used. This section is indispensable in order for
the readers to understand what has been modified in the
transmission procedures of DSTBCs in our proposed ASTs.
It is also vital for the readers to notice the underlying re-
quirement of all conventional DSTBCs that the channel
coefficients must be constant during at least two consec-
utive code blocks. We also show here in which scenar-
ios DSTBCs (differential detection) should be used instead
of STBCs (coherent detection).
2.1. Conventional DSTBCs without diversity antenna
selection
Differential space-time block codes are the candidate for the
channels where fading changes so fast that the transmission
of the training signals (eg., a large overhead) is either im-
practical or uneconomical. DSTBCs have been considered
intensively and a number of DSTBCs have been proposed
in literature such as [11–17]. In [2, 3], we have proved
that all conventional DSTBCs (without antenna selection)
provide a full spatial diversity order.
Let us consider the unitary DSTBC proposed by Ganesan
et al. in [13] as an example. We consider a system with
NTx antennas and KRx antennas. Let Rt,A,Ntbe the
(K×N)-sized matrices of received signals at time t, chan-
nel coefficients between Rx and Tx antennas, and noise at
the Rx antennas, respectively. The
κη
th element of A,
namely a
κη
, is the channel coefficient of the path between
the
η
th Tx antenna and the
κ
th Rx antenna. Channel co-
efficients are assumed to be identically independently dis-
tributed (i.i.d.) complex, zero-mean Gaussian random vari-
ables. Noises are assumed to be i.i.d. complex Gaussian
random variables with the distribution CN(0,
σ
2).
Let {sj}p
j=1={sR
j+isI
j}p
j=1(where i2=−1,sR
jand sI
jare
the real and imaginary parts of sj, respectively) be the set
of psymbols, which are derived from a unitary power sig-
nal constellation Sand transmitted in the tth block. Conse-
quently, each symbol has a unitary energy, i.e., |sj|2=1.
We define a matrix Zt=1
√p∑p
j=1(XjsR
j+iYjsI
j), where the
square, order-Nweighting matrices {Xj}p
j=1and {Yj}p
j=1
are orthogonal themselves and they satisfy the permutation
property. These weighting matrices are considered as the
amicable orthogonal designs (AODs). The backgrounds
on AODs can be found in [22]. The coefficient 1
√pis to
guarantee that Ztis a unitary matrix, i.e., ZtZH
t=I.
For illustration, the Alamouti DSTBC corresponding to
N=2is defined as
Zt=1
√2s1s2
−s∗
2s∗
1.(1)
A DSTBC corresponding to N=4is given below:
Zt=1
√3
s1s2s30
−s∗
2s∗
10s3
−s∗
30s∗
1−s2
0−s∗
3s∗
2s1
.(2)
The transmission starts with an initial, identity, order-Nma-
trix W0=INcarrying no information. The matrix trans-
mitted at time t(t=1,2,3...)is given by
Wt=Wt−1Zt.(3)
As Ztis a unitary matrix, the matrix Wtis also a unitary
one. The model of the channel at time t, for t=0,1,2...,
(t=0means the transmission of the first block W0, i.e., the
initial transmission) is:
Rt=AWt+Nt.(4)
In all propositions of conventional DSTBCs, the channel
coefficients must be constant during at least two adjacent
code blocks2, i.e., constant during at least 2Nsymbol time
slots (STSs). It means that if the channel coefficient ma-
trix Ais assumed to be constant over two consecutive
blocks t−1and t, the maximum likelihood (ML) detector
for the symbols {sj}p
j=1is calculated as follows [13, 23]:
{ˆsj}p
j=1=Argmax
{sj},sj∈S
Re{tr(RH
tRt−1Zt)},(5)
where Arg{.}denotes the argument operation, tr(.)denotes
the trace operation, Re{.}and Im{.}denote the real and
the imaginary parts of the argument, respectively.
If we denote Tcto be the average coherent time of the chan-
nel which represents the time-varying nature of the chan-
nel, then the channel is considered to be constant during
2This means that the channel coefficients are constant during each win-
dow of at least two consecutive code blocks and windows do not overlap
each other.
80
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
this time. Therefore, after each duration Tc, the transmit-
ter restarts the transmission and transmits a new initial
block W0followed by other code blocks Wt(t=1,2,3...).
These procedures are repeated until all data are transmitted.
Due to the orthogonality of DSTBCs, the transmitted sym-
bols are decoded separately, rather than jointly. Therefore,
if we denote:
Dj=Retr(RH
tRt−1Xj)+iRetr(RH
tRt−1iYj)(6)
then the ML detector for the symbol sjis [2, 3]:
ˆsj=Argmax
sj∈SRe{D∗
jsj},(7)
where D∗
jis the conjugate of Dj.
Expressions (6) and (7) show that the detection of the sym-
bol sjis carried out without the knowledge of channel co-
efficients. Particularly, the symbol sjcan be decoded by
using the received signal blocks in the two consecutive
transmission times, provided that the channel coefficients
are constant during two consecutive code blocks (otherwise,
we will not have the decoding expressions (5) and (7)).
The requirement that the channel coefficients must be con-
stant during at least 2 consecutive code blocks can be re-
laxed if the linear prediction is used at the receiver. In this
scenario, the receiver uses multiple previous received code
blocks Rt−1,Rt−2,etc., to predict the relation between the
current channel coefficient matrix, say At, and the previ-
ous channel coefficient matrices. This approach has been
mentioned in [24]. Certainly, the penalty of this approach
is the complexity of the receiver structure.
It has been proved in our paper [2, 3] that all conven-
tional DSTBCs (without ASTs) provide a full diversity of
order NK , where Nand Kare the number of Tx and Rx
antennas, respectively. We also can realize this observation
in Section 4 of this paper.
2.2. Remarks on the time-varying Rayleigh fading
channels
According to the frequency of channel coefficient changes,
we distinguish three typical scenarios which are usually
examined in practice and present the most common, real
propagation conditions (see [17, p. 13] and [25, p. 2]).
1. Channel coefficient matrix Ais random and its en-
tries change randomly at the beginning of each sym-
bol time slot (STS) and are constant during one STS.
This scenario is referred to as the fast Rayleigh flat
fading channel.
2. Ais random and its entries change randomly after
a duration containing a number of STSs. This sce-
nario is referred to as the block Rayleigh flat fading
channel. The example of this scenario will be men-
tioned later.
3. Ais random but is selected at the beginning of trans-
mission and its entries keep constant all the time.
This scenario is referred to as the slow or quasi-static
Rayleigh flat fading channel. Local area networks
(LANs) or wide local area networks (WLANs) with
a slow fading rate and a high data rate are the exam-
ples of the quasi-static Rayleigh flat fading channels,
where the channel coefficients may be constant dur-
ing thousands of STSs.
Given the above clarifications, we have the following impor-
tant note. Owing to the condition that channel coefficients
must be constant during, at least, two consecutive code
blocks, in all conventional DSTBCs mentioned in litera-
ture, the channels are considered as block fading channels,
although the coherent time of the channels in the case of
DSTBCs (with differential detection) are much shorter than
that in the case of STBCs (with coherent detection).
To illustrate, for the case of the Alamouti DSTBC, the chan-
nel coefficients must be constant during at least 4 STSs.
During the first two STSs, the initial, order-2, identity ma-
trix I2which carries no information is transmitted. During
the next two STSs, the Alamouti code carrying 2 symbols
is transmitted. This note clarifies how fast fading channels
may change when DSTBCs is utilized. Certainly, a longer
coherent duration of the channel results in a more efficient
utilization of DSTBCs.
We give 2 examples of block Rayleigh fading channels
where coherent STBCs or DSTBCs can be used.
Example 1: We consider the scenario where the Alamouti
STBCs with coherent detection can be used for the cellular
mobile system with the carrier frequency Fc=900 MHz.
Speed of the mobile user is v=5km/h (walking speed)
and the STS is assumed to be Ts=0.125 ms (equivalently,
the baud rate is Fs=8KBd/s. Denote c=3.108m/s to
be light speed. The maximum Doppler frequency is then
calculated as
fm=vFc/c=4.17 Hz .
The average coherent time Tcof the channel is estimated
by the following empirical expression [26, p. 204]:
Tc=0.423
fm
=101.52 ms .
It means that the channel coefficients can be considered to
be constant during almost Tc/Ts≈812 consecutive STSs,
i.e., approximately 406 consecutive Alamouti code blocks.
In this case, the channel coefficients change so slow that the
training signals can be transmitted. In other words, STBCs
with coherent detection are preferred than DSTBCs with
differential detection.
Example 2: We consider another scenario where the
Alamouti DSTBCs with different detection can be used
for the cellular mobile system with the carrier frequency
Fc=900 MHz. Speed of the mobile user is v=60 km/h
(vehicular speed) and the STS is assumed to be Ts=0.5ms
corresponding to the baud rate Fs=2KBd/s. The maxi-
mum Doppler frequency is then calculated as
fm=vFc/c=50 Hz .
81
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
Similarly, the average coherent time Tcof the channel is
estimated as [26, p. 204]
Tc=0.423
fm
=8.46 ms .
It means that the channel coefficients can be consid-
ered to be constant during Tc/Ts≈16 consecutive STSs,
i.e., 8 consecutive Alamouti code blocks. The channel is
a block Rayleigh fading one where DSTBCs can be em-
ployed. In this case, it is either impractical or uneconomical
to use STBCs with coherent detection since the coherent
time is too short to transmit multiple training symbols in
order for the receiver to estimate the channel coefficients.
3. Definitions, notations
and assumptions
For ease of exposition, we define some notations as follows.
Definition 1: ̥is defined as an order-Noperation on M
non-negative, real numbers {
ε
1,...,
ε
M}where the Nin-
dices (N<M) corresponding to the Nlargest values out of
Mvalues {
ε
1,...,
ε
M}are selected. We denote this opera-
tion as ̥N(
ε
1,...,
ε
M). The output of the operation ̥is
the set of Nindices which is denoted by ˆ
IN.
Example 3: M=3,N=2,
ε
1=10,
ε
2=20 and
ε
3=30.
We have:
ˆ
I2=̥2(
ε
1,
ε
2,
ε
3) = {2,3}.
The elements of the set ˆ
I2are the indices of
ε
2and
ε
3,
which are in turns the 2 largest values among {
ε
1,
ε
2,
ε
3}.
Definition 2: We define the (M,N;K,L)AST/DSTBC
scheme to be the transmitter and receiver diversity antenna
selection technique for channels using DSTBCs with dif-
ferential detection where NTx antennas are selected out of
MTx antennas (N<M), while LTx antennas are selected
out of KRx antennas (L<K) for transmission.
Given that notation, the (M,N;K)AST/DSTBC scheme
refers to as the transmitter diversity antenna selection tech-
nique for channels using DSTBCs with differential detec-
tion where NTx antennas are selected out of MTx anten-
nas (N<M) for transmission. All KRx antennas are used
without selection. Similarly, the (M;K,L)AST/DSTBC
scheme refers to as the receiver diversity AST for chan-
nels using DSTBCs where LTx antennas are selected out
of KRx antennas for transmission, while MTx antennas are
used without selection.
In the paper, we mainly focus on the transmitter diver-
sity AST, i.e., the (M,N;K)AST/DSTBC schemes. We
sometimes compare the proposed (M,N;K)AST/DSTBC
schemes with the respective schemes in channels which use
STBCs with coherent detection. Hence, similarly, we use
the notation (M,N;K)AST/STBC to refer to the transmit-
ter diversity AST for channels using STBCs with coherent
detection.
For example, if M=4,N=2and K=1, then the (4,2;1)
AST/DSTBC is the AST where the 2Tx antennas are se-
lected (depending on certain criteria) from 4Tx antennas
for transmission, while the receiver has 1Rx antenna.
Some assumptions considered in the paper are given below.
Assumption 1: The channel coefficients between the trans-
mitter and receiver antennas are assumed to be i.i.d. com-
plex, zero-mean Gaussian random variables. Noises are
assumed to be i.i.d. complex Gaussian random variables
with the distribution CN(0,
σ
2). These assumptions are
applicable when the Tx and Rx antennas are sufficiently
separated from one another (by a multiple of half of the
wavelength) so that the Tx (and Rx) antennas are uncorre-
lated. The scenario where the antennas are correlated will
be examined in our future works.
Assumption 2: Although channels with differential detec-
tion change faster than those with coherent detection, so
that the transmission of multiple training signals is uneco-
nomical (and, consequently, the utilization of DSTBCs is
useful), we make a reasonable assumption that it is possi-
ble to transmit a few feedback bits (for each channel co-
herent duration Tc) from the receiver to the transmitter via
a feedback channel with a certain feedback error rate. The
feedback error rate is typically assumed to be 4% to 10%.
Finally, we want to stress the following important remarks.
Remark 1: Due to the tiny size of the receivers, such as the
hand-held mobile phones in the cellular mobile systems, it
is well known that employment of more than 2Tx anten-
nas at the receiver is uneconomical. Hence, the receiver
diversity antenna selection is not considered in this paper,
although the generalization of the proposed techniques for
the receiver diversity antenna selection is straightforward.
Remark 2: We use the modified notation (N+¯
N,N;K)
AST/DSTBC, rather than (M,N;K)AST/DSTBC, where
M=N+¯
N, to refer to our 3th proposed AST/DSTBC
scheme in this paper. The main purpose of using this nota-
tion is to stress that ¯
NTx antennas among (N+¯
N)available
Tx antennas are the standby Tx antennas. These standby
Tx antennas are only used in certain conditions stipulated
by the selection criteria. Those selection criteria will be
mentioned in more details later.
4. Basis of transmitter antenna selection
for channels using DSTBCs
In our papers [2, 3], we have proved that all conventional
DSTBCs mentioned in literature, such as [13–16], pro-
vide a full spatial diversity order. This means that, if
the channel contains NTx and KRx antennas, then square,
order-NDSTBCs provide a full spatial diversity of or-
der NK provided that the DSTBCs have a full rank.
Let us consider the unitary DSTBCs mentioned in Sec-
tion 2.1 for instance. It is shown in [2, Eq. (11)], [3, Eq. (9)],
82
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
[12] and [23, Eq. (5.30)], that the SNR of the statistic Dj
in Eq. (6) is approximately:
SNRd i f f ≈kAk2
F
2p
σ
2
=tr(AHA)
2p
σ
2
=
N
∑
η
=1K
∑
κ
=1|a
κη
|2
2p
σ
2,(8)
where kAkFis the Frobenius norm of the matrix A. Clearly,
SNR has 2N K freedom degrees. As a result, the unitary
DSTBC considered provides a full spatial diversity of or-
der NK .
Let
ξη
≡∑K
κ
=1|a
κη
|2(
η
=1,...,N) be the total power of
signals received by KRx antennas during each STS. We can
rewrite SNRd i f f as follows:
SNRd i f f ≈
N
∑
η
=1
ξη
2p
σ
2.(9)
It is obvious that greater values of
ξη
s result in a greater
SNRd i f f .
Let us consider a system comprising MTx antennas
(M>N) and KRx antennas. We now want to select
the Nbest Tx antennas out of MTx antennas so that
SNRd i f f is maximized. From Eqs. (8) or (9), to maximize
SNRd i f f , we need to maximize kAk2
F. Equivalently, the N
first maximum values out of Mvalues {
ξ
1,
ξ
2,...,
ξ
M}=
∑K
κ
=1|a
κ
1|2,∑K
κ
=1|a
κ
2|2,...,∑K
κ
=1|a
κ
M|2must be se-
lected. In other words, the indices of the Nbest Tx antennas
are selected by the following antenna selection criterion:
ˆ
IN=̥N
ξ
1,...,
ξ
M
=̥NK
∑
κ
=1|a
κ
1|2,
K
∑
κ
=1|a
κ
2|2,...,
K
∑
κ
=1|a
κ
M|2.(10)
Again, note that the transmitter diversity antenna selection,
rather than receiver diversity antenna selection, is examined
in this paper. All KRx antennas are used without antenna
selection.
The selection criterion in Eq. (10) is applicable only when
the channel coefficients are perfectly known at the receiver.
This scenario is realistic when the channel changes so
slowly that the multiple training signals can be transmit-
ted. This scenario is commonly examined in channels us-
ing STBCs with coherent detection. The ASTs are referred
to as the (M,N;K)AST/STBC schemes which have been
intensively considered in literature [5–10].
As oppose to coherent detection, in channels using DSTBCs
with differential detection, channel coefficients change
faster so that the transmission of multiple training signals is
either impractical or uneconomical, and consequently, the
channel coefficients are unknown at the receiver.
Therefore, the antenna selection criterion in Eq. (10) cannot
be directly applied to channels using DSTBCs with differ-
ential detection. However, we will show that this criterion
can be modified to apply to channels using DSTBCs with
differential detection.
Particularly, we will prove later in this paper that, at high
SNRs, the statistical properties, i.e., means and variances,
of the received signals r0
κη
s – the elements of the matrix
R0received during the initial transmission – are similar
to those of the channel coefficients a
κη
s. As a result, at
high SNRs, maximizing kR0k2
Ftends to be the same as
maximizing kAk2
F.
Based on this observation, we propose the modified an-
tenna selection scheme for channels using DSTBCs. The
transmitter selects Tx antennas on the basis of the compar-
ison, which is carried out once per each channel coherent
duration Tcat the receiver, between the power of the signals
which are received by all KRx antennas during the initial
transmission (the first block W0).
If we denote ˆ
INto be the set of the Nindices of the Tx an-
tennas which should be selected, then the modified antenna
selection criterion for channels using DSTBCs is:
ˆ
IN=̥N
χ
1,...,
χ
M
=̥NK
∑
κ
=1|r0
κ
1|2,
K
∑
κ
=1|r0
κ
2|2,...,
K
∑
κ
=1|r0
κ
M|2.
This modified selection criterion is mentioned in more
details in the so-called general (M,N;K)AST/DSTBC
scheme proposed as below.
5. The general (M,N;K)AST/DSTBC
for channels utilizing DSTBCs
In this section, we generalize our AST/DSTBC proposed
in [2, 3] for channels using DSTBCs with arbitrary numbers
of Tx and Rx antennas.
Let us consider a system containing MTx antennas and
KRx antennas using the unitary, square, order-NDSTBCs
(N<M) proposed by Ganesan et al. [13, 27]. Note that
the proposed ASTs are also applicable to any conventional
DSTBC regardless of being unitary or not.
In the following analysis, the normal, lower case letters
denote scalars, the bold, lower case letters denote vectors,
while the bold upper case letters denote matrices. For sim-
plicity, we omit the superscripts indicating the different co-
herent durations Tcs of the channel when a certain coher-
ent duration is being considered. The superscripts are only
used when we consider different coherent durations Tcs si-
multaneously.
The general (M,N;K)AST/DSTBC is proposed as follows:
•At the beginning of transmission, the transmitter
sends an initial block ˜
W0=IMvia MTx antennas,
rather than sending an initial block W0=INvia
83
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
NTx antennas like in all conventional DSTBCs. This
transmission is referred to as the initial transmission.
We note the change in the size of matrices compared
to Eq. (4) by using the tilde mark for matrices as
below:
˜
W0=IM,
˜
A=a1a2... aM,
˜
N0=n01 n02 ... n0M ,
where aj(j=1...M) is the column vector of the
channel coefficients ai j (i=1...K) corresponding to
the channel from the jth Tx antenna to the ith Rx an-
tenna, i.e, aj=[a1j,...,aK j]T, and n0j is the noise af-
fecting these channels during the initial transmission,
i.e., n0j = [n01 j,...,n0K j]T. Here, the superscript T
denotes the transposition operation.
•The receiver determines the matrix ˜
R0of received
signals during the initial transmission as given below:
˜
R0=˜
A˜
W0+˜
N0
=˜
AIM+˜
N0
=r01 r02 ... r0M (11)
=a1+n01 a2+n02 ... aM+n0M ,
where
r0j =aj+n0j
= [a1j+n01 j,...,aK j +n0K j ]Tj=1...M.
•From the initial received matrix ˜
R0, the receiver de-
termines semiblindly the Nbest channels based on
the initial, received matrix ˜
R0by comparing Mterms
χ
j=kr0jk2
F, for j=1...M, i.e., comparing the total
power of the signals received by all KRx antennas
from the jth Tx antenna during the jth STS:
χ
j=
K
∑
i=1|r0i j|2=
K
∑
i=1|ai j +n0i j |2(12)
to search for the first Nmaximum values. In other
words, the antenna selection criterion is:
ˆ
IN=̥N
χ
1,...,
χ
M
=̥NK
∑
i=1|r0i1|2,
K
∑
i=1|r0i2|2,...,
K
∑
i=1|r0iM|2
=̥NK
∑
i=1|ai1+n0i1|2,
K
∑
i=1|ai2+n0i2|2,...,
K
∑
i=1|aiM +n0iM|2,(13)
where ˆ
INdenotes the set of Nindices of the Tx an-
tennas which should be selected.
Without loss of generality, we assume here that these
maximum values are corresponding to the first Nel-
ements in the matrix ˜
R0, i.e.,
ˆ
IN={1,2,...,N}.
Then, the receiver carries out the two following tasks:
1. The receiver informs the transmitter via a feed-
back channel to select the first NTx antennas to
transmit data.
2. The receiver generates the matrix R0, which
is used to decode the next code blocks, by
taking the first Nelements of the matrix ˜
R0,
corresponding to the first Nmaximum values,
i.e., R0=a1+n01 a2+n02 ... aN+n0N .
•The transmitter selects the NTx antennas indicated
by the feedback information. In this case, the first
NTx antennas are selected to transmit data. The trans-
mission is now exactly the same as that in the system
using the Nfirst Tx antennas only.
If Tcis the average coherent time of the channel, then after
each duration Tc, the transmitter restarts the transmission
and transmits a new initial block ˜
W0followed by other
code blocks Wt(t=1,2,3...). The above procedures are
repeated until all data are transmitted.
The transmission procedure is shown in Fig. 1. The su-
perscripts are used to indicate the different coherent dura-
tions Tcs of the channel. The code blocks ˜
W0are transmit-
ted via MTx antennas in MSTSs and the following blocks
via NTx antennas in NSTSs.
Fig. 1. Transmission of DSTBCs without (a) and with (b) the
antenna selection technique.
From the aforementioned algorithm, we have following re-
marks.
Remark 3: At the transmitter, after the initial matrix ˜
W0
is transmitted, the next matrices Wt(t=1,2,3,...) can
be calculated by using a tacit default matrix W0=IN
in Eq. (3). We use the term tacit default matrix to refer to
84
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
the fact that the matrix W0=INis tacitly used at the trans-
mitter to generate the next code blocks Wtby Eq. (3), rather
than being actually transmitted. Owing to this fact, it is also
important to note that the generation of the matrices Wt
does not necessarily take place after the transmitter obtains
the feedback information. Instead, the next code blocks Wt
are automatically generated by multiplying the previous
block Wt−1with the tacit default matrix W0=INfollow-
ing Eq. (3).
Remark 4: The above proposed AST is carried out with
only Ntraining = (M−N)training symbols for each coher-
ent duration Tc. The typical values of Ntraining are 1 or
2 symbols.
Remark 5: The number of feedback bits required to in-
form the transmitter about the best channels in the general
(M,N;K)AST/DSTBC is:
N=log2M
N,(14)
where ⌈.⌉is the ceiling function.
Remark 6: In all conventional DSTBCs, the initial matrix
W0=INis only used to initialize the transmission. Partic-
ularly, W0is used to calculate the next transmitted matrices
following Eq. (3), and to generate the initial, received ma-
trix R0directly, which is combined with the next receiving
matrix R1to decode transmitted symbols.
Unlike the conventional DSTBCs without ASTs, in the pro-
posed technique, the initial identity matrix ˜
W0=IMis
transmitted. This matrix has two main roles. It enables
the receiver to generate the initial, received matrix R0in-
directly (from the received matrix ˜
R0). Simultaneously, in
some sense, it also plays a role of training signals, which
assist the receiver to determine semiblindly the best chan-
nels. This is the main difference between the differential
space-time coding with our AST and the one without AST.
Remark 7: Similarly to the conventional DSTBCs without
ASTs mentioned in Section 3, in our proposed technique,
channel coefficients are required to be constant during at
least two consecutive code blocks. Therefore, the channels
must be constant during, at least, (M+N)STSs in our pro-
posed AST, while they must be unchanged during at least
2NSTSs in all conventional DSTBC techniques without the
proposed ASTs if the delay of transmitting feedback infor-
mation from the receiver to the transmitter is not consid-
ered. In the case when the delay is considered, the channel
coefficients must stay longer.
Remark 8: The procedures of the proposed general
(M,N;K)AST/DSTBC is more explicitly presented in
Fig. 2. Steps 1a, 1b, 4 and 5 are carried out at the transmit-
ter, while the remaining steps are carried out at the receiver.
As stated earlier, Step 1b is not necessarily carried out af-
ter Step 3a finishes. In other words, the transmitter can
perform Step 1b right after finishing Step 1a. Similarly,
because the matrix R0is created straightforwardly from
Fig. 2. The general (M,N;K)AST/DSTBC for the system using
DSTBCs.
the matrix ˜
R0, the receiver can perform Step 3b right af-
ter finishing Step 3a. These properties reduce unnecessary
delays during transmission.
6. The restricted (M,N;K)AST/DSTBC
and the (N+¯
N,N;K)AST/DSTBC
As mentioned in Eq. (14), the number of feedback bits
required in the general (M,N;K)AST/DSTBC is:
N=log2M
N.
It is easy to realize that, Nis large for large values of M
and N. For instance, in the general (6,4;K) AST/DSTBC
(Kis arbitrary), we have N=4. Therefore, it is either im-
practical or uneconomical to employ the general (M,N;K)
AST/DSTBC for large values of Mand N, except when
either the number of feedback bits or the time required to
process feedback information is reduced.
Motivated by this observation, we derive here the two
AST/DSTBC schemes which are the modifications of the
aforementioned, general (M,N;K)AST/DSTBC scheme.
We refer those ASTs to as the restricted (M,N;K)
AST/DSTBC and the (N+¯
N,N;K)AST/DSTBC. The two
modified ASTs are more amenable to practical implementa-
tion in channels using DSTBCs than the general (M,N;K)
AST/DSTBC.
The restricted (M,N;K)AST/DSTBC requires only 1 feed-
back bit, while providing a relatively good bit error per-
formance. Meanwhile, the (N+¯
N,N;K)AST/DSTBC re-
quires at most an equal number of feedback bits as
the general (M,N;K)AST/DSTBC where M=N+¯
N,
while shortening the time required to process feedback
information. Especially, when ¯
N=1, the (N+1,N;K)
AST/DSTBC scheme provides the same bit error per-
formance as the general (M,N;K)AST/DSTBC scheme,
where M=N+1, while shortening the processing time
for feedback information. For ¯
N>1, there exists a degra-
85
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
dation of the bit error performance of the (N+¯
N,N;K)
AST/DSTBC scheme, compared to the general (M,N;K)
AST/DSTBC scheme where M=N+¯
N. Therefore, the
(N+1,N;K)AST/DSTBC scheme is of our particular in-
terest in this paper.
6.1. The restricted (M,N;K) AST/DSTBC
In the scenario where the capacity limitation of the feed-
back channel, especially in the uplink channels of the 3G
mobile communication systems, needs to be considered,
the number of feedback bits is as small as possible. More
importantly, limiting the number of feedback bits is nec-
essary when fading changes fast. Based on the general
(M,N;K)AST/DSTBC mentioned in Section 5, we propose
here the restricted (M,N;K)AST/DSTBC for channels us-
ing DSTBCs, where only 1 feedback bit is required for each
channel coherent duration Tcto inform the transmitter.
In the restricted (M,N;K)AST/DSTBC, the set of MTx
antennas is divided into two subsets. Each subset includes
NTx (N<M) antennas. Subsets may partially overlap each
other. Figure 3 presents 3 cases for illustration. In Fig. 3a,
Fig. 3. Some examples of the transmitter antenna grouping
for (a) the restricted (4,2;K) AST/DSTBC; (b) the restricted
(3,2;K) AST/DSTBC; (c) the restricted (5,4;K) AST/DSTBC.
we give an example where 4Tx antennas are divided into
2 subsets including 2Tx antennas each, while in Fig. 3b,
3Tx antennas are divided into 2 subsets containing
2Tx each. These 2 cases can be applied, for instance, to the
Alamouti DSTBC with the restricted (4,2;K) AST/DSTBC
and with the restricted (3,2;K) AST/DSTBC, respectively.
Figure 3c, we derive other example where 5Tx antennas
are divided into 2 subsets which partially overlap one an-
other and include 4Tx antennas each. This case can be
applied, for instance, to the order-4 DSTBC with the re-
stricted (5,4;K) AST/DSTBC.
Let Ψand Φbe the sets of indices indicating the order of
the Tx antennas in each subset, respectively. The selec-
tion criterion for the restricted (M,N;K)AST/DSTBC is
as follows.
During each coherent duration Tcof the channel, the re-
ceiver compares:
∑
j∈Ψ
χ
j=∑
j∈ΨK
∑
i=1|r0i j|2
and
∑
j∈Φ
χ
j=∑
j∈ΦK
∑
i=1|r0i j|2,
ie., the receiver compares the total power of the signals
received by all KRx antennas during the initial transmis-
sion from two subsets of Tx antennas, and then informs the
transmitter to select the subset providing the greater total
power. If ∑j∈Ψ
χ
jis lager, then the receiver, via a feed-
back loop, informs the transmitter to select the Tx anten-
nas corresponding to the set of indices Ψ. Otherwise, the
Tx antennas corresponding to the set of indices Φshould
be selected. These procedures are repeated for different co-
herent durations Tcs of the channel until the transmission
of data is completed.
It is obvious that only one feedback bit per each coherent
time Tcis required for transmission diversity purpose.
6.2. The (N
N
N+¯
N
N
N,N
N
N;K
K
K)AST/DSTBC
In this section, we consider a system containing M=
(N+¯
N)Tx antennas and KRx antennas and transmitting
square, order-NDSTBCs. Among MTx antennas, NTx an-
tennas are called default Tx antennas which are normally
used to transmit signals, and ¯
Nremaining Tx antennas are
the standby ones which are only used when the selection
criterion is satisfactory. The diagram of the system in this
technique is shown in Fig. 4.
Fig. 4. The diagram of the (N+¯
N,N;K) AST/DSTBC.
86
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
We propose here a modified AST/DSTBC scheme for this
structure of the system which is referred to as the (N+
¯
N,N;K)AST/DSTBC. This AST shortens the time required
to process feedback information in comparison with the
general (M,N,K)AST/DSTBC where M=N+¯
N.
Note that ¯
Nis strictly smaller than N, i.e., ¯
N<N. It will
be shown later that when ¯
N=N, the (N+¯
N,N;K)
AST/DSTBC turns into the restricted (M,N;K)
AST/DSTBC where M=N+¯
N.
Without loss of generality, we number (N+¯
N)Tx antennas
by indices from 1 to (N+¯
N), and assume that the Ndefault
Tx antennas are indexed from 1 to Nwhile the ¯
Nstandby
Tx antennas are indexed from (N+1)to (N+¯
N).
Similarly to the general (M,N,K)AST/DSTBC, in the
(N+¯
N,N;K)AST/DSTBC, the transmitter starts the
transmission by transmitting an identity, order-Mmatrices
˜
W0=IM=IN+¯
Nduring each channel coherent time Tc.
Let ˜
R0be the initial, received matrix ˜
R0during the initial
transmission, i.e., the time when the initial matrix ˜
W0is
transmitted. Similarly to Eq. (11), we have:
˜
R0=r01 r02 ... r0N r0N+1.. . r0N+¯
N.
In this expression, r0j is the column vector of the signals
received by all KRx antennas during the jth STS from the
jth Tx antenna. Let
χ
j=kr0jk2
Fwhich is the total power
received by all KRx antennas from the jth Tx antenna
(j=1,...,N+¯
N).
We denote
ϕ
kto be the set of ¯
Nindices of the ¯
Ndefault
Tx antennas which are arbitrarily taken from Ndefault
Tx antennas. There are total q=N
¯
Nsuch sets. Further-
more, for k=1,...,q, we denote:
α
k=∑
j∈
ϕ
k
χ
j
=∑
j∈
ϕ
kkr0jk2
F
=∑
j∈
ϕ
kK
∑
i=1|r0i j|2.
The proposed (N+¯
N,N;K)AST/DSTBC is as follows. On
the one hand, the receiver searches for the minimum value
among qvalues {
α
1,...,
α
q}. Let
α
be this minimum value
and ˆ
I¯
Nbe the set of indices of the corresponding default
Tx antennas. This action can be mathematically presented
by
α
=min
α
1,...,
α
q.
On the other hand, the receiver calculates the total power
of the received signals value which are received by
all KRx antennas during the initial transmission from
¯
Nstandby Tx antennas. If we denote this total power to
be
β
, then this action can be expressed as:
β
=
(N+¯
N)
∑
j=(N+1)
χ
j
=
(N+¯
N)
∑
j=(N+1)kr0jk2
F
=
(N+¯
N)
∑
j=(N+1)K
∑
i=1|r0i j|2.
If
α
≥
β
, then the Tx antennas which the transmitter should
select are all default Tx antennas {1,...,N}.
If
α
<
β
, the ¯
Ndefault Tx antennas which have the indices
listed in the set ˆ
I¯
Nwill be replaced by the standby antennas.
To illustrate, we assume that ˆ
I¯
N={1,2,..., ¯
N}, i.e., the
first ¯
Ndefault Tx antennas provide the minimum value
α
.
If
α
≥
β
, then the Tx antennas are {1,2,...,N}. Oth-
erwise, the first ¯
Ndefault Tx antennas are replaced by
the ¯
Nstandby Tx antennas. Consequently, the NTx an-
tennas which should be selected are {¯
N+1, .. . , N−1,N,
N+1,...,N+¯
N}.
The antenna selection mechanism for this example is pre-
sented more clearly by the flowchart in Fig. 5.
Fig. 5. The flow chart of the (N+¯
N,N;K) AST/DSTBC .
Associated with this antenna selection mechanism, we pro-
pose the structure of the feedback information as presented
in Fig. 6. The bit Blis used to indicate whether the trans-
Fig. 6. The proposed structure of the feedback information for
channels using DSTBCs.
87
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
mitter has to replace ¯
Ndefault antennas with the standby
ones. The bit Blis zero if the answer is no, i.e.,
α
≥
β
,
and Blis unity otherwise. The lfollowing bits indicate
which ¯
Nantennas among Ndefault antennas should be re-
placed by the standby ones.
It is easy to realize that l=log2N
¯
N. With this struc-
ture, the transmitter first considers the bit Bl. As soon
as it realizes that Bl=0, the rest of the feedback infor-
mation is not necessarily processed. The transmitter will
transmit signals via the default Tx antennas {1,2, ... , N}.
If Bl=1, the transmitter uses the lfollowing bits
Bl−1, .. . , B0to recognize which default antennas should
be replaced by the standby ones.
Therefore, the number of feedback bits required to be trans-
mitted in the (N+¯
N,N;K)AST/DSTBC is at most equal
to:
N1=l+1=1+log2N
¯
N.(15)
We want to stress that, theoretically, there is no need to
transmit lbits Bl−1, .. . , B0in the case Bl=0. If so, a sin-
gle feedback bit (bit Bl) is required to be transmitted.
Note that the number of feedback bits required to be trans-
mitted and processed in the general (M,N;K)AST/DSTBC
where M=N+¯
Nis always:
N2=log2M
N=log2N+¯
N
N.
It is easy to realize that if Nis a power of 2, we have
N1≤N2. For instance, for N=2and ¯
N=1, we have
N1=N2=2. For N=4and ¯
N=2, we have N1=3and
N2=4.
Therefore, if Nis the power of 2, the number of feed-
back bits required to be transmitted in the (N+¯
N,N;K)
AST/DSTBC is almost equal to that required in the gen-
eral (M,N;K)AST/DSTBC (M=N+¯
N). The number of
feedback bits required to be processed in the (N+¯
N,N;K)
AST/DSTBC is either (l+1), which is equal to the num-
ber of transmitted feedback bits N1, or only 1 (smaller
than N1) depending on the bit Bl. The smaller number of
feedback bits required to be transmitted and to be processed
in the (N+¯
N,N;K)AST/DSTBC shortens the time re-
quired to process feedback information in the (N+¯
N,N;K)
AST/DSTBC in comparison with the time required in the
general (M,N;K)AST/DSTBC. The quantitative estima-
tion of this time reduction will be mentioned later.
From the aforementioned algorithm, we have the following
remarks on the (N+¯
N,N;K)AST/DSTBC.
Remark 9: Theoretically, it is not necessary to transmit
lbits Bl−1, .. ., B0in the case Bl=0. Only one feedback
bit Blis required to be transmitted (and processed) in this
case. This observation may further shortens the time for
feeding information back.
Remark 10: The Ndefault Tx antennas are always used
for transmission whenever
β
≤
α
, i.e., the set of ¯
Nstandby
antennas is not better3than the worst set of ¯
Ndefault Tx an-
tennas among Ndefault Tx antennas.
When
β
>
α
, i.e, the set of ¯
Nstandby antennas is better
than the worst set of ¯
Ndefault Tx antennas among Ndefault
Tx antennas, these ¯
Nstandby antennas are used to replace
the ¯
Ndefault antennas.
Remark 11: If ¯
N=N, not only the antenna selection cri-
terion of the (N+¯
N,N;K)AST/DSTBC is exactly the
same as that of the restricted (M,N;K)AST/DSTBC where
M=N+¯
N, but the required numbers of feedback bits
of both ASTs are also the same (only 1 feedback bit
is required). Therefore, the (N+¯
N,N;K)AST/DSTBC
turns into the restricted (M,N;K)AST/DSTBC. Owing
to this reason, ¯
Nmust be strictly smaller than Nin the
(N+¯
N,N;K)AST/DSTBC.
Remark 12: If 2≤¯
N<N, the (N+¯
N,N;K)AST/DSTBC
is suboptimal as the set containing the Nbest Tx anten-
nas among (N+¯
N)Tx antennas is not always selected
for transmission, and consequently, it provides a worse
BER performance than the general (M,N;K)AST/DSTBC
where M=N+¯
N. In return for this disadvantage, the
(N+¯
N,N;K)AST/DSTBC shortens the time required to
process feedback information in comparison with the gen-
eral (M,N;K)AST/DSTBC.
Remark 13: If ¯
N=1, the antenna selection criterion of the
(N+¯
N,N;K)AST/DSTBC turns into the selection criterion
of the general (M,N;K)AST/DSTBC where M=N+¯
N=
N+1. Intuitively, both the (N+1,N;K)AST/DSTBC
and the general (M,N;K)AST/DSTBC select the Nopti-
mal Tx antennas out of (N+1)Tx antennas. Consequently,
the BER performance of the (N+1,N;K)AST/DSTBC
is the same as that of the general (M,N;K)AST/DSTBC
(M=N+1).
The main advantage of the (N+1,N;K)AST/DSTBC over
the general (M,N;K)AST/DSTBC is that the time required
to process feedback information in the former is shorter
than that in the later. This advantage will be mentioned in
more details in the next section in which the quantitative
estimation of the time reduction gained by the (N+1,N;K)
AST/DSTBC in comparison with the general (M,N;K)
AST/DSTBC is derived.
Owing to these reasons, the (N+¯
N,N;K)AST/DSTBC
with ¯
N=1, i.e., the (N+1,N;K) AST/DSTBC, is of our
particular interest in this paper.
Let
χ
j=∑K
i=1|r0i j|2for j=1,...,(N+1). The (N+1,
N;K) AST/DSTBC scheme can be slightly modified from
the (N+¯
N,N;K)AST/DSTBC and stated as follows.
The receiver searches for the minimum value
χ
min among
(N+1)values
χ
1,...,
χ
N+1, i.e.:
χ
min =min
χ
1,...,
χ
N+1.
We assume that
χ
min ≡
χ
nwhere n=1,...,(N+1).
3A better set provides a larger total power which is received by all
KRx antennas during the initial transmission.
88
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
If n≡(N+1), then all Ndefault Tx antennas are used
to transmit signals. In this case, bit Bl=0. Otherwise,
the indexed-ndefault Tx antenna is replaced by the standby
Tx antenna (the (N+1)th Tx antenna). This standby an-
tenna is combined with the (N−1)Tx antennas to transmit
signals. In this case, bit Bl=1.
7. Relative reduction of the average
processing time of the (N+¯
N,N;K)
AST/DSTBC
In order to estimate the time reduction obtained by the
(N+¯
N,N;K)AST/DSTBC, we compare the average time
required to process feedback information in this AST and
that required in the general (M,N;K)AST/DSTBC (M=
N+¯
N) in Section 5.
Although, there is a fact that the time required to process the
feedback information does not necessarily increase linearly
with the number of feedback bits, it is easier to calculate
the time benefit of the proposed technique when the aver-
age processing time is assumed to increase linearly with the
number of feedback bits. Obviously, the result we derive as
follows is only aimed at providing the readers with the lower
bound of the relative reduction of the average processing
time obtained by the (N+¯
N,N;K)AST/DSTBC in com-
parison with that of the general (M,N;K)AST/DSTBC.
Let P
0be the probability of the event that the set of
¯
Nstandby Tx antennas is not used in the (N+¯
N,N;K)
AST/DSTBC. In other words, P
0is the probability of the
event that
β
≤
α
, i.e. P
0=P(
β
≤
α
). Similarly, let P
1be
the probability of the event that the ¯
Nstandby Tx antennas
are used for transmission, i.e, P
1=P(
β
>
α
). Clearly, we
have P
1= (1−P
0).
We now calculate P
0in the two following cases which are
different in the underlying essences.
•When ¯
N=1, as mentioned earlier in Remark 13, the
default Tx antenna is only used when it is the worst
Tx antenna among (N+1)Tx antennas. We make
a reasonable assumption that the event where a cer-
tain Tx antenna (either default or standby antenna)
is the worst antenna among (N+1)Tx antennas is
equiprobable. Then we have:
P
0=P(
β
≤
α
) = 1
N+1
1=1
(N+1).(16)
•When ¯
N≥2, we make a reasonable assumption that
the event in which a set containing the certain ¯
N
default Tx antennas selected from the Navailable
default Tx antennas is the worst set, is equiprobable.
This means that:
P(
α
≡
α
1) = ···=P(
α
≡
α
q) = 1
N
¯
N=1
q.
We also assume the following conditional probability:
P(
β
≤
α
|
α
≡
α
k) = 0.5
for k=1,...,q. As a result, we have:
P
0=P(
β
≤
α
)
=
q
∑
k=1
P(
β
≤
α
|
α
≡
α
k)P(
α
≡
α
k)
=
q
∑
k=1
0.5·1
q
=0.5.(17)
Let
ϑ
be the average processing time for 1 feedback bit. Be-
cause the transmitter has to process 1 feedback bit (bit Bl)
only if
β
≤
α
and has to process N1=1+log2N
¯
N
feedback bits if
β
>
α
, the average time required to process
feedback information in the (N+¯
N,N;K)AST/DSTBC is:
τ
1=P
0
ϑ
+P
1N1
ϑ
=P
0
ϑ
+ (1−P
0)1+log2N
¯
N
ϑ
.
On the other hand, in the general (M,N;K)AST/DSTBC
where M=N+¯
N, the transmitter always has to process
N2=log2N+¯
N
N feedback bits. Therefore, the average
processing time is:
τ
2=N2
ϑ
=log2N+¯
N
N
ϑ
.
Hence, the relative reduction of the average processing time
between two techniques is:
△
τ
τ
2
∆
=
τ
2−
τ
1
τ
2
=1−1+ (1−P
0)log2N
¯
N
log2N+¯
N
¯
N .(18)
For ¯
N=1, from Eqs. (16) and (18), we have:
△
τ
τ
2
=1−1+ (1−1
N+1)log2N
log2(N+1).
For ¯
N≥2, from Eqs. (17) and (18), we have:
△
τ
τ
2
=1−1+0.5log2N
¯
N
log2N+¯
N
¯
N .
The relative time reduction △
τ
τ
2[%] for some particular
values of Nand ¯
Nis presented by the table in Fig. 7. We
only need to calculate the time reduction for the pair of N
and ¯
Nsatisfying ¯
N<N.
From this table, we realize that the average processing time
reduction is considerable even for ¯
N=1. In this case,
the average processing time reduction for N=2, 4 and 8
is 16.67, 13.33 and 8.33%, respectively. To illustrate, the
(2 + 1, 2; 1) AST/DSTBC in the system using the Alam-
outi DSTBC with 2 default Tx antennas, 1 standby Tx an-
tenna and 1Rx antenna gains the relative time reduction
of 16.67%.
89
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
Fig. 7. Relative time reduction [%] of the (N+¯
N,N;K)
AST/DSTBC compared to the general (M,N;K)AST/DSTBC
where M=N+¯
N.
It is worth to stress that the time reduction is probably
much greater than the above figures if we take its non-
linear proportionality with the number of feedback bits into
consideration.
8. Some comments on spatial diversity
order of the proposed ASTs
In this section, we consider the spatial diversity order of
the ASTs proposed for channels using DSTBCs with dif-
ferential detection. To do that, at first, we review the same
issue for channels using STBCs with coherent detection, to
provide the readers with the state of the art of this issue.
The spatial diversity order of the ASTs for channels using
space-time codes with coherent detection has been some-
what examined in a few papers, such as [8–10, 28–31].
Particularly, in [28] and [30], the authors considered the
combination of the transmitter antenna selection and space-
time trellis codes (STTCs) and proved that the (M,2;1)
AST/STTC and (M,2;2) AST/STTC schemes provide
a full spatial diversity order when SNR is very large
(see Eqs. (26) and (27) in [28]) as long as the STTCs have
a full rank. In [31], the authors considered the receiver
(not transmitter) diversity selection associated with the use
of STCs (either STBCs or STTCs) in MIMO systems over
the quasi-static (slow) Rayleigh fading channels. The au-
thor proved there that the (M;K,L)AST/STC schemes
(where MTx antennas are used without selection, while
the Lbest Tx antennas are selected out of KRx antennas)
provide a full spatial diversity order of MK, provided that
the STCs have a full rank (see Eq. (10) in [31]).
It is noted that, in this paper, we consider the transmitter
(not receiver ) diversity antenna selection and the use of
DSTBCs which have orthogonal structures. Therefore, it
is useful to review the spatial diversity order of the ASTs
associated with STBCs only (not STTCs or other STCs).
Having this note in mind, we realize that there are very few
works, such as [10] and [29], have mentioned the spatial
diversity order of transmitter diversity ASTs for channels
using STBCs. In [10] and [29], the authors limited them-
selves to consider the Alamouti STBC modulated by a bi-
nary phase shift keying (BPSK) signal constellation in the
(M,2;1) AST/STBC and (M,2;2) AST/STBC schemes only.
Those studies are far from the exhaustive research.
In other words, the exhaustive research on the spatial di-
versity order of transmitter diversity ASTs is still missing
even for space-time coded systems with coherent detection.
For space-time coded systems with non-coherent detection,
such as the systems using DSTBCs, the study on the spa-
tial diversity order of AST/DSTBC schemes has not been
examined yet. Due to this reason, in this paper, we do not
have ambition to examine this issue for all cases, which
certainly requires a lot of studies in future.
Instead, we show that the problem of finding the spatial
diversity order of the ASTs proposed for channels using
DSTBCs with differential detection is the same as that prob-
lem for the case of coherent detection when SNR ≫1. Once
this has been shown, we consider the (M,2;1) AST/DSTBC
and (M,2;2) AST/DSTBC schemes. Since the respective
(M,2;1) AST/STBC and (M,2;2) AST/STBC schemes for
channels using STBCs provide a full spatial diversity or-
der [10, 29], then the (M,2;1) AST/DSTBC and (M,2;2)
AST/DSTBC schemes for channels using DSTBCs also
provide a full spatial diversity order as if all Tx and Rx an-
tennas were used.
We restrict ourselves to consider only the general (M,N;K)
AST/DSTBC scheme for illustration. Other schemes, such
as the restricted (M,N;K)AST/DSTBC scheme or the
(N+¯
N,N;K)AST/DSTBC scheme are similarly analyzed.
To begin with, we review some crucial discussions men-
tioned in [10] on the spatial diversity order achieved
by the (M,2;K) AST/STBC schemes for channels using
STBCs with coherent detection. We use the superscript l
(l=1,2,3,...) to indicate the different coherent durations
of the channel. Since the coherent detection is being con-
sidered, the channel coefficients between Tx and Rx anten-
nas denoted by ¯a(l)
i j , for i=1,...,Kand j=1,...,M, are
assumed to be perfectly known at the receiver and partially
known at the transmitter through a feedback channel. Let
¯
ξ
(l)
j=∑K
i=1|¯a(l)
i j |2. We assume that ¯a(l)
i j s are i.i.d. complex
Gaussian random variables with the distribution CN(0,
σ
a).
With the notation mentioned in Section 3 of this paper,
we rewrite the Tx antenna selection criterion, which was
mentioned by Eq. (1) in [10], for the (M,2;K) AST/STBC
scheme during the lth coherent duration as
ˆ
I(l)
2=̥2¯
ξ
(l)
1,¯
ξ
(l)
2,..., ¯
ξ
(l)
M
=̥2K
∑
i=1|¯a(l)
i1|2,
K
∑
i=1|¯a(l)
i2|2,...,
K
∑
i=1|¯a(l)
iM|2.(19)
90
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
Denote
γ
=Eb
N0to be the SNR per bit. It has been shown
in [10], the BER expression, say P
2,1, of the (M,2;1)
AST/STBC, where there is only 1Rx antenna, in flat
Rayleigh fading channels for binary phase shift keying mod-
ulation asymptotically approaches (see Eq. (7) in [10]):
P
2,1≈(2M−1)!
22M−1(M−1)!1
γ
M
when
γ
→∞. This equation shows that a full diversity order
of Mis achieved asymptotically for the (M,2;1) AST/STBC
when
γ
→∞.
The BER expression, say P
2,2, of the (M,2;2) AST/STBC,
where there are 2Tx antennas, in flat Rayleigh fading chan-
nels for BPSK modulation asymptotically approaches (see
Eq. (8) in [10]):
P
2,2≈M(4M−1)!
25M−2(2M−1)(2M−1)!1
γ
2M
when
γ
→∞. This equation shows that a full diversity
order of 2Mis achieved asymptotically for the (M,2;2)
AST/STBC when
γ
→∞.
The cases for K≥3are not practically significant since
it is difficult to employ more than 2Tx antennas at the
mobile set in mobile communication downlinks. Due to this
reason, the cases for K≥3were not presented in [10].
Now we return to consider our proposed, general (M,2;K)
AST/DSTBC for channels using DSTBCs with differential
detection. The superscript k(k=1,2,3,...,m) is used to
indicate the different coherent durations of the channel (see
Fig. 1). Since the differential detection is considered, the
channel coefficients between Tx and Rx antennas a(k)
i j , for
i=1,...,K,j=1,...,M,k=1,...,m, are unknown at
either the receiver or the transmitter.
As mentioned in Eq. (13) in Section 5, the selection cri-
terion for the general (M,2;K) AST/DSTBC during the
kth coherent duration is:
ˆ
I(k)
2=̥2
χ
(k)
1,...,
χ
(k)
M
=̥2K
∑
i=1|r(k)
0i1|2,
K
∑
i=1|r(k)
0i2|2,...,
K
∑
i=1|r(k)
0iM|2
=̥2K
∑
i=1|a(k)
i1+n(k)
0i1|2,
K
∑
i=1|a(k)
i2+n(k)
0i2|2,...,
K
∑
i=1|a(k)
iM +n(k)
0iM|2.(20)
We assume that the channel coefficients a(k)
i j s and noise
n(k)
0i js are i.i.d. complex Gaussian random variables with
the distribution CN(0,
σ
a)and CN(0,
σ
), respectively. We
consider the mean and the variance of the following term:
µ
(k)
i j
∆
=|a(k)
i j +n(k)
0i j|2
for i=1,...,K,j=1,...,Mand k=1,...,m.
Since a(k)
i j and n(k)
0i j are the i.i.d. zero-mean, complex Gaus-
sian random variables, (a(k)
i j +n(k)
0i j)are the i.i.d., complex
Gaussian random variables with the distribution CN(0,
ρ
)
where
ρ
=
σ
a+
σ
. Therefore,
µ
(k)
i j are the i.i.d, central chi-
squared random variables with n=2degrees of freedom
and with the following mean and variance [32, p. 42]:
E{
µ
(k)
i j }=n
ρ
2=
ρ
,
σµ
(k)
i j
=2n
ρ
22
=
ρ
2.
We investigate the case in which the channel SNR ≫1.
Equivalently, the variances of noise terms n(k)
0i js are very
small in comparison with the variances of a(k)
i j s, and there-
fore,
ρ
≈
σ
a. As a result, the means and the variances
of
µ
(k)
i j are approximately:
E{
µ
(k)
i j } ≈
σ
a
σµ
(k)
i j ≈
σ
2
a,(21)
when SNR ≫1.
On the other hand, we consider the following term:
θ
(k)
i j =|a(k)
i j |2
for i=1,...,K,j=1,...,Mand k=1,...,m.
Similarly analyzed,
θ
(k)
i j are the i.i.d, central chi-squared
random variables having n=2degrees of freedom with
the following mean and variance [32, p. 42]:
E{
θ
(k)
i j }=
σ
a,
σµ
(k)
i j
=
σ
2
a.(22)
From Eqs. (21) and (22), we realize that,
µ
(k)
i j s and
θ
(k)
i j s
have the same statistical properties, i.e., means and vari-
ances when SNR ≫1. We can rewrite the antenna selection
criterion of the (M,N;K)AST/DSTBC in Eq. (20) as
ˆ
I(k)
2≈̥2K
∑
i=1|a(k)
i1|2,
K
∑
i=1|a(k)
i2|2,...,
K
∑
i=1|a(k)
iM |2(23)
when SNR ≫1.
Clearly, the antenna selection criterion for the (M,2;K)
AST/DSTBC scheme now tends to be the same as the cri-
terion mentioned in Eq. (19) for the (M,2;K) AST/STBC
scheme.
We may conclude that, if the channel SN R →∞, the be-
havior of the (M,2;K) AST/DSTBC scheme proposed for
channels using DSTBCs with differential detection tends to
be the same as that of the (M,2;K) AST/STBC scheme men-
tioned in literature for channels using STBCs with coher-
ent detection, although the (M,2;K) AST/DSTBC scheme
is inferior by 3 dB compared to the (M,2;K) AST/STBC
scheme due to the fact that the channel coefficients are not
91
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
known at either transmitter or receiver. As a result, because
the (M,2;1) AST/STBC and (M,2;2) AST/STBC schemes
achieve a full spatial diversity [10, 29], then so do the
(M,2;1) AST/DSTBC and (M,2;2) AST/DSTBC schemes,
provided that the channel SNR is very large.
9. Simulation results
In this section, we run some Monte-Carlo simulations to
solidify our proposed AST/DSTBC schemes. We consider
a wireless link comprising K=1Rx antenna. The chan-
nel SNR is defined to be the ratio between the total aver-
age power of the received signals and the average power
of noise at the Rx antenna during each STS. Note that the
numbers of feedback bits which are required for the general
(M,N;K)AST/DSTBC and the (N+¯
N,N;K)AST/DSTBC
examined in the simulations are calculated by Eqs. (14)
and (15), respectively. The number of feedback bit required
for the restricted (M,N;K)AST/DSTBC is always 1. In
simulations, DSTBCs are modulated by a QPSK signal con-
stellation in simulations.
First, the Alamouti DSTBC in Eq. (1) corresponding to
N=2is simulated. We consider 4 following scenarios:
– Alamouti DSTBC without ASTs;
– Alamouti DSTBC with the general (3,2;1)
AST/DSTBC (2 feedback bits);
– Alamouti DSTBC with the restricted (3,2;1)
AST/DSTBC (1 feedback bit);
– Alamouti DSTBC with the (2+1,2;1) AST/DSTBC
(N=2,¯
N=1, 2 feedback bits).
However, as noted earlier in Remark 13 of Section 6.2, the
(2+1,2;1) AST/DSTBC has the same BER performance as
the general (3,2;1) AST/DSTBC, although the time required
to process feedback information in the former is shorter
than that in the later. For this reason, we do not need to
plot the BER performance of the (2+1,2;1) AST/DSTBC
scheme.
Furthermore, in each AST/DSTBC scheme, we exam-
ine 2 cases where the feedback error rates are assumed
to be 4% and 10%. Transmit antennas in the restricted
(3,2;1) AST/DSTBC are grouped by the scheme mentioned
in Fig. 3b.
Note that it would be better if we can compare the per-
formances here with the performance of a DSTBC with-
out ASTs which provides the same spatial diversity or-
der as the diversity order (equal to 3) provided by the
proposed AST/DSTBC schemes, i.e., the general (3,2;1)
AST/DSTBC, the restricted (3,2;1) AST/DSTBC and the
(2+1,2;1) AST/DSTBC. This means that we should com-
pare the performance of the Alamouti DSTBC (associated
with the proposed ASTs) with that of an order-3 DSTBC
(without ASTs). However, while the Alamouti DSTBC has
a full rate, it is well known that DSTBCs of an order be-
ing greater than 2 with a full rate do not exist. For this
reason, it is unfair to compare the Alamouti DSTBC with
an order-3 DSTBC, because they have different code rates,
and consequently, we do not plot the performance of any
order-3 DSTBC in the simulation.
As analyzed earlier, channel coefficients must be constant
during at least two adjacent code blocks. If Tcdenotes the
coherent time of the channel, then it is required that:
•Tc≥4STSs for the Alamouti DSTBC without ASTs;
•Tc≥5STSs for the Alamouti DSTBC with the gen-
eral (3,2;1) AST/DSTBC, with the restricted (3,2;1)
AST/DSTBC, or with the (2+1,2;1) AST/DSTBC.
Therefore, to compare fairly the performance of the Alam-
outi DSTBC with different ASTs, the simulation is run for
Tcwhich is not less than 5 STSs. Example 2 in Section 2.2
is one of such practical scenarios.
The performance of the Alamouti DSTBC with and with-
out ASTs is shown in Fig. 8. It can be seen from Fig. 8
Fig. 8. The Alamouti DSTBC with the general (3,2;1)
AST/DSTBC and the restricted (3,2;1) AST/DSTBC.
that the proposed ASTs significantly improve the BER per-
formance of the channel. Again, the BER performances
of the (2+1,2;1) AST/DSTBC is exactly the same as that
of the general (3,2;1) AST/DSTBC. The main advantage
of the (2+1,2;1) AST/DSTBC over the general (3,2;1)
AST/DSTBC is that the time required to process feedback
information is shortened by 16.67% (see Fig. 7). The SNR
reductions [dB] gained by our proposed ASTs to achieve
the same BER =10−3as the Alamouti DSTBC without
ASTs are given in Table 1.
Next, we consider the general (4,2;1) AST/DSTBC (3 feed-
back bits) and the restricted (4,2;1) AST/DSTBC (1 feed-
92
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
back bit) in which the transmitter selects N=2Tx antennas
out of M=4Tx antennas. Clearly, in this case, we have
¯
N=M−N=2, i.e., ¯
N=N. In Remark 11, we have stated
that the (2+2,2;1) AST/DSTBC reduces to the restricted
(4,2;1) AST/DSTBC. Therefore, we do not plot the per-
formance of the (2+2,2;1) AST/DSTBC here. Transmitter
antennas in the restricted (4,2;1) AST/DSTBC are grouped
by the scheme mentioned in Fig. 3a.
Table 1
SNR reductions [dB] of the general (3,2;1) AST/DSTBC,
the restricted (3,2;1) AST/DSTBC and the (2+1,2;1)
AST/DSTBC in the channel using Alamouti DSTBC
Error General (3,2;1) (2+1,2;1) Restricted (3,2;1)
[%]AST/DSTBC AST/DSTBC AST/DSTBC
4 3.25 3.25 2.9
10 2.25 2.25 2.25
Similarly, it is required that:
•Tc≥4STSs for the Alamouti DSTBC without ASTs;
•Tc≥6STSs for the Alamouti DSTBC with the
general (4,2;1) AST/DSTBC or with the restricted
(4,2;1) AST/DSTBC.
To compare fairly the performance of Alamouti DSTBC
with different ASTs, the simulation is run for Tcwhich is
not less than 6 STSs. Example 2 mentioned in Section 2.2
is one of such practical scenarios.
Fig. 9. The Alamouti DSTBC with the general (4,2;1)
AST/DSTBC and the restricted (4,2;1) AST/DSTBC schemes.
The performance of the proposed AST/DSTBC schemes is
presented in Fig. 9. The SNR reductions [dB] achieved by
our proposed ASTs to have the same BER =10−3as the
DSTBC without ASTs are given in Table 2.
Table 2
SNR reductions [dB] of the general (4,2;1) AST/DSTBC
and the restricted (4,2;1) AST/DSTBC in the channel
using Alamouti DSTBC
Error General (4,2;1) Restricted (4,2;1)
[%]AST/DSTBC AST/DSTBC
4 3.5 4.3
10 1.5 3.25
Finally, we examine the square, order-4, unitary DSTBC in
Eq. (2) corresponding to N=4and the code rate 3/4. We
consider the following 4 scenarios:
– DSTBC without ASTs;
– DSTBC with the general (5,4;1) AST/DSTBC
(3 feedback bits);
– DSTBC with the restricted (5,4;1) AST/DSTBC
(1 feedback bit);
– DSTBC with the (4+1,4;1) AST/DSTBC (N=4,
¯
N=1, 3 feedback bits). Similarly, the BER per-
formance of the (4+1,4;1) AST/DSTBC is exactly
the same as that of the general (5,4;1) AST/DSTBC,
and therefore, we do not need to plot the BER per-
formance of the (4+1,4;1) AST/DSTBC in the simu-
lation.
In each AST, we also consider 2 cases where the feedback
error rates are assumed to be 4% and 10%. Transmitter
antennas in the restricted (5,4;1) AST/DSTBC are grouped
by the scheme mentioned in Fig. 3c.
It is required that:
•Tc≥8STSs for DSTM without ASTs;
•Tc≥9STSs for DSTM with the general
(5,4;1) AST/DSTBC, with the restricted (5,4;1)
AST/DSTBC or with the (4+1,4;1) AST/DSTBC.
Therefore, the simulation is run for Tcwhich is not less
than 9 STSs. Example 2 in Section 2.2 is still valid for this
scenario.
The performance of the proposed AST/DSTBC schemes
is presented in Fig. 10. It is noted that the (4+1,4;1)
AST/DSTBC provides the same BER performance as that
of the general (5,4;1) AST/DSTBC (see Remark 13 in Sec-
tion 6.2), while shortening the time which is required to
process feedback information by 13.33% (see Fig. 7) com-
pared to the general (5,4;1) AST/DSTBC.
The SNR reductions [dB] achieved by our proposed ASTs
to have the same BER =10−3as the DSTBC without ASTs
are given in Table 3.
From all the above simulations, we realize that the proposed
ASTs significantly improve the performance of wireless
channels using DSTBCs. Also, we realize that the restricted
93
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
Fig. 10. Square, order-4, unitary DSTBC with the general (5,4;1)
AST/DSTBC and the restricted (5,4;1) AST/DSTBC schemes.
(M,N;K)AST/DSTBC provide a relatively good BER per-
formance compared to the general (M,N;K)AST/DSTBC
and the (N+¯
N,N;K)AST/DSTBC, while requiring only
1 feedback bit. More importantly, the restricted (M,N;K)
AST/DSTBC may perform even better than the general
(M,N;K)AST/DSTBC and the (N+¯
N,N;K)AST/DSTBC
when the feedback error rate grows large. Intuitively, this
Table 3
SNR reductions [dB] of the proposed (5,4;1)
AST/DSTBCs in the channel using square, order-4,
unitary DSTBC
Error General (5,4;1) (4+1,4;1) Restricted (5,4;1)
[%]AST/DSTBC AST/DSTBC AST/DSTBC
4 1.2 1.2 1
10 0.8 0.8 0.85
is interpreted by the fact that the restricted AST requires
only 1 feedback bit while the remaining ASTs require mul-
tiple feedback bits. Therefore, when the feedback error
rate grows large, the feedback information in the restricted
ASTs is less likely erroneous than that in the other ASTs.
As a result, the restricted ASTs are the practical candidates
for the channels where fading changes fast.
10. Discussions and conclusion
In this paper, we propose three ASTs referred to as the
general (M,N;K)AST/DSTBC, the restricted (M,N;K)
AST/DSTBC, and the (N+¯
N,N;K)AST/DSTBC for
the channels using DSTBCs with arbitrary number of Tx
and Rx antennas.
Since the general (M,N;K)AST/DSTBC scheme requires
a large number of feedback bits when M,Nand Kare large,
it is either impractical or uneconomical for implementa-
tion in such cases. The restricted (M,N;K)AST/DSTBC
and the (N+¯
N,N;K)AST/DSTBC schemes overcome this
shortcoming.
Particularly, the restricted (M,N;K)AST/DSTBC is an
attractive technique, which provides relatively good bit
error performance, compared to the general (M,N;K)
AST/DSTBC, while requiring only 1 feedback bit. This
advantage is very important in the case where the capacity
limitation of the feedback channel, such as in the uplink
channels of the 3G mobile communication systems, is con-
sidered. This advantage is also very beneficial in the chan-
nels where fading changes fast and/or the feedback error
rate in the feedback channel grows large.
Unlike the restricted AST/DSTBC schemes, where we try
to reduce the number of feedback bits, in the (N+¯
N,N;K)
AST/DSTBC schemes, we reduce the average time required
to process feedback information. These techniques use at
most the same number of feedback bits and provide the
same BER performance (if ¯
N=1) as that of the general
(M,N;K)AST/DSTBC schemes (M=N+¯
N), but remark-
ably reduce the average time required to process feedback
information.
Simulation show that all three proposed ASTs with a limited
number (typically, 1 or 2) of training symbols per each co-
herent duration of the channel noticeably improve the BER
performance of wireless channels utilizing DSTBCs. The
improvement is significant even for the case of 1 training
symbol, i.e., in the general (M,N;K)AST/DSTBC where
M= (N+1); in the restricted (M,N;K)AST/DSTBC
where M= (N+1); or in the (N+1,N;K) AST/DSTBC
schemes.
The restricted (M,N;K)AST/DSTBC may provide a better
BER performance over the general (M,N;K)AST/DSTBC
and the (N+¯
N,N;K) AST/DSTBC when the feedback error
rate is large. Hence, the restricted AST/DSTBC schemes
are a good choice for the channels where fading changes
fast and/or the feedback error rate is large.
It is noted that, in this paper, we assume that the carrier
phase/frequency is perfectly recovered at the receiver. In
fact, phase/frequency recovery errors may exist, which de-
grade the performance of the proposed ASTs. Those er-
rors may occur due to the difference between the frequency
of the local oscillators at the transmitter and the receiver,
and/or due to the Doppler frequency-shift effect. The ef-
fect of imperfect carrier recovery on the performance of the
proposed ASTs in wireless channels utilizing DSTBCs has
been examined in our paper [4]. Readers may refer to [4]
for more details.
Also, in this paper, the delay of feedback information has
not been considered. In reality, the delay of feedback in-
formation may somewhat degrade the overall performance
of the proposed ASTs. This issue will be mentioned in our
94
Transmitter diversity antenna selection techniques for wireless channels utilizing differential space-time block codes
other works. Finally, as mentioned earlier, the exhaustive
research on the spatial diversity order of the ASTs proposed
for channel using DSTBCs has not been derived yet and it
must be fully examined in the future work.
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Le Chung Tran received the
excellent B.Eng. degree with
the highest distinction and the
M.Eng. degree with the high-
est distinction in telecommuni-
cations engineering from Hanoi
University of Communications
and Transport, and Hanoi Uni-
versity of Technology, Vietnam,
in 1997 and 2000, respectively.
From March 2002 to July 2005,
he studied towards the Ph.D. degree in telecommunica-
tions engineering at the School of Electrical, Computer and
Telecommunications Engineering, University of Wollon-
gong, Australia. He is currently working as Associate Re-
search Fellow at the Telecommunications and Information
Technology Research Institute (TITR), School of Electri-
cal, Computer and Telecommunications Engineering, Uni-
versity of Wollongong, Australia. He has been working
as a lecturer at Hanoi University of Communications and
Transport, Vietnam, since September 1997 to date. He has
achieved numerous national and overseas awards, includ-
ing WUS – World University Services (twice), Vietnamese
Government’s Scholarship, UPA – Wollongong Univer-
sity Postgraduate Award, Wollongong University Tuition
Fee Waver, and Alexander von Humboldt (AvH) Research
95
Le Chung Tran, Tadeusz A. Wysocki, Alfred Mertins, and Jennifer Seberry
Fellowship during the undergraduate and postgraduate time.
His research interests include transmission diversity tech-
niques, mobile communications, space-time processing,
MIMO systems, channel propagation modelling, ultra-wide
band communications, OFDM, and spread spectrum tech-
niques.
e-mail: lct71@uow.edu.au
University of Wollongong
Northfields Avenue
Wollongong, NSW 2522, Australia
Tadeusz Antoni Wysocki re-
ceived the M.Sc.E. degree
with the highest distinction in
telecommunications from the
Academy of Technology and
Agriculture, Bydgoszcz, Po-
land, in 1981. In 1984, he re-
ceived his Ph.D. degree, and in
1990, was awarded a D.Sc. de-
gree (habilitation) in telecom-
munications from the Warsaw
University of Technology. In 1992, Doctor Wysocki moved
to Perth, Western Australia to work at Edith Cowan Univer-
sity. He spent the whole 1993 at the University of Hagen,
Germany, within the framework of Alexander von Hum-
boldt Research Fellowship. After returning to Australia, he
was appointed a Program Leader, Wireless Systems, within
Cooperative Research Centre for Broadband Telecommu-
nications and Networking. Since December 1998 he has
been working as an Associate Professor at the University of
Wollongong, NSW, within the School of Electrical, Com-
puter and Telecommunications Engineering. The main ar-
eas of Doctor Wysocki’s research interests include: indoor
propagation of microwaves, code division multiple access
(CDMA), space-time coding and MIMO systems, as well
as mobile data protocols including those for ad hoc net-
works. He is the author or co-author of four books, over
150 research publications and nine patents. He is a Senior
Member of IEEE.
wysocki@uow.edu.au
University of Wollongong
Northfileds Ave
Wollongong, NSW 2522, Australia
Alfred Mertins received his
Dipl.-Ing. degree from the Uni-
versity of Paderborn, Germany,
in 1984, the Dr.-Ing. degree
in electrical engineering and
the Dr.-Ing. habil. degree in
telecommunications from the
Hamburg University of Tech-
nology, Germany, in 1991 and
1994, respectively. From 1986
to 1991 he was with the Hamburg University of Technol-
ogy, Germany, from 1991 to 1995 with the Microelectron-
ics Applications Center Hamburg, Germany, from 1996 to
1997 with the University of Kiel, Germany, from 1997 to
1998 with the University of Western Australia, and from
1998 to 2003 with the University of Wollongong, Australia.
In April 2003, he joined the University of Oldenburg, Ger-
many, where he is a Professor in the Faculty of Mathematics
and Science. His research interests include speech, audio,
image and video processing, wavelets and filter banks, and
digital communications.
e-mail: alfred.mertins@uni-oldenburg.de
University of Oldenburg
Signal Processing Group
Institute of Physics
26111 Oldenburg, Germany
Jennifer Seberry graduated
Ph.D. in computation mathe-
matics from La Trobe Univer-
sity in 1971. She has sub-
sequently held positions at
the Australian National Univer-
sity, The University of Syd-
ney and ADFA, The Univer-
sity of New South Wales. She
has published extensively in dis-
crete mathematics and is world
renown for her new discoveries on Hadamard matrices
and statistical designs. In 1970 she co-founded the series
of conferences known as the 20th Australian Conference
on Combinatorial Mathematics and Combinatorial Com-
puting. She started teaching in cryptology and computer
security in 1980. She is especially interested in authenti-
cation and privacy. In 1987, at University College, ADFA,
she founded the Centre for Computer and Communications
Security Research to be a reservoir of expertise for the Aus-
tralian community. Her studies of the application of dis-
crete mathematics and combinatorial computing via bent
functions, S-box design, has led to the design of secure
crypto-algorithms and strong hashing algorithms for secure
and reliable information transfer in networks and telecom-
munications. Her studies of Hadamard matrices and or-
thogonal designs are applied in CDMA technologies. In
1990 she founded the AUSCRYPT/ASIACRYPT series of
International Cryptologic Conferences in the Asia/Oceania
area. She has supervised 25 successful Ph.D. candidates,
has over 350 scholarly papers and six books.
e-mail: jennie@uow.edu.au
University of Wollongong
Northfields Avenue
Wollongong, NSW 2522, Australia
96
