Differential coding for non-orthogonal space-time block codes with non-unitary constellations over arbitrarily correlated rayleigh channels

Abstract

In this paper, we propose a maximum likelihood (ML) decoder for differentially encoded full-rank square nonorthogonal space-time block codes (STBCs) using unitary or non-unitary signal constellations, which is also applicable to full-ranked orthogonal STBC (OSTBC). As the receiver is jointly optimized with respect to the channel and the unknown data, it does not require any knowledge of channel power, signal power, or noise power to decode the signal, and the decision is purely based on two consecutively received data blocks. We analyze the effect of channel correlation on the performance of the proposed system in Rayleigh fading channels. Assuming a general correlation model, an upper bound of the pair-wise error probability (PEP) of the differential OSTBCs is derived. An approximate bound of the PEP for the differential nonorthogonal STBCs is also derived. We propose a precoder designing criterion for differential STBC over arbitrarily correlated Rayleigh channels. Precoding improves the system performance over the correlated Rayleigh MIMO channels. Our precoded differential codes differ from the previously proposed precoder designs for differential OSTBC in the following ways: 1) We propose a precoder design for arbitrarily correlated Rayleigh channels, whereas the previous work considers only for transmit correlation. 2) The previous work is only applicable to the OSTBCs with PSK constellations, whereas our precoder is applicable to any type of full-rank square STBCs with unitary and non-unitary signal constellations.

Full text

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Differential Coding for Non-Orthogonal Space-Time
Block Codes with Non-Unitary Constellations over
Arbitrarily Correlated Rayleigh Channels
Manav R. Bhatnagar, Member, IEEE, Are Hjørungnes, Senior Member, IEEE,
and Lingyang Song, Member, IEEE
Abstract— In this paper, we propose a maximum likelihood
(ML) decoder for differentially encoded full-rank square non-
orthogonal space-time block codes (STBCs) using unitary or
non-unitary signal constellations, which is also applicable to full-
ranked orthogonal STBC (OSTBC). As the receiver is jointly
optimized with respect to the channel and the unknown data, it
does not require any knowledge of channel power, signal power,
or noise power to decode the signal, and the decision is purely
based on two consecutively received data blocks. We analyze the
effect of channel correlation on the performance of the proposed
system in Rayleigh fading channels. Assuming a general corre-
lation model, an upper bound of the pair-wise error probabil-
ity (PEP) of the differential OSTBCs is derived. An approximate
bound of the PEP for the differential non-orthogonal STBCs
is also derived. We propose a precoder designing criterion for
differential STBC over arbitrarily correlated Rayleigh channels.
Precoding improves the system performance over the correlated
Rayleigh MIMO channels. Our precoded differential codes differ
from the previously proposed precoder designs for differential
OSTBC in the following ways: 1) We propose a precoder design
for arbitrarily correlated Rayleigh channels, whereas the previous
work considers only for transmit correlation. 2) The previous
work is only applicable to the OSTBCs with PSK constellations,
whereas our precoder is applicable to any type of full-rank square
STBCs with unitary and non-unitary signal constellations.
Index Terms: Differential space-time codes, Non-unitary
constellations, Precoded differential codes, Non-orthogonal
codes.
I. INTRODUCTION
Multiple-input multiple-output (MIMO) communications
systems have drawn a lot of attention [1]–[3]. They provide
high capacity and diversity gains as compared to single-
input single-output (SISO) systems and, therefore, MIMO
systems are suitable for the high data-rates required by the
next generation wireless communication systems. In MIMO
systems, space-time block codes (STBCs) are used to pro-
vide the maximum diversity [1]–[3]. However, the coherent
detection of STBC requires perfect channel state information
(CSI), which is gained by transmission of training data at the
cost of reduced effective data-rate. Furthermore, this method
Corresponding Author: Manav R. Bhatnagar is with UNIK – University
Graduate Center, University of Oslo, Instituttveien 25, P. O. Box 70, NO-2027
Kjeller, Norway, email: manav@unik.no.
Are Hjørungnes is with UNIK – University Graduate Center, University
of Oslo, Instituttveien 25, P. O. Box 70, NO-2027 Kjeller, Norway, email:
arehj@unik.no.
Lingyang Song is with School of Electronics Engineering and Computer
Science, Peking University, No.5 Yiheyuan Road Haidian District, Beijing,
P.R.China 100871, email: lingyang.song@pku.edu.cn.
This work was supported by the Research Council of Norway project
176773/S10 called OptiMO, which belongs to the VERDIKT program.
is suitable only for the channels that remain constant for
several symbol durations. If the channel remains constant
for small number of symbol durations, this method does not
work well, since the channel estimates might be very wrong
when it is going to be used by the receiver. Differential
STBC may be used to overcome this difficulty and improve
the data-rate. In [4]–[6], the differential coding for square
orthogonal STBCs (OSTBCs) with unitary constellations like
QPSK, which have all the points on a circle, is suggested.
These differential OSTBCs have a simple encoding structure
and their decoding does not need knowledge of the channel
coefficients. However, this differential system does not work
with square non-orthogonal high data-rate STBCs [7]–[11] and
non-unitary constellations like QAM. Whereas, high data-rate
STBC and QAM constellations are very useful in providing
higher data-rates.
In [12], [13], differential codes for non-unitary constel-
lations are suggested but they are applicable to the square
orthogonal space-time structures and rectangular QAM con-
stellations only. The differential codes, suggested for non-
unitary constellations [14], [15] are also applicable to square
orthogonal STBCs only and their decoding requires knowl-
edge of channel power, signal power, and noise power. In [14],
the channel power is estimated by buffering many samples
(normally L ≥ 100) of the received data and then finding
the average autocorrelation of the received data sequence.
These methods introduce a delay in making decisions and the
performance of the receiver may degrade in the fastly varying
channels, which do not remain constant for many symbol
durations. All these existing differential schemes cannot be
applied to the full-rank square non-orthogonal STBCs like
high-rate STBCs, which use optimized non-unitary constel-
lations for maximizing the coding gain [7]–[9] or achieving
the diversity multiplexing trade-offs [10], [11]. High rate
STBCs are very useful for improving the performance of the
wireless communications systems in the terms of data-rates
and capacity.
In this paper, our main contributions are as follows:
1) We derive an ML decoder for differential STBCs with
non-unitary constellations like QAM, which follow the full-
rank square non-orthogonal structures like high-rate STBCs.
The proposed decoder also provides better performance for
OSTBCs as compared to the previously suggested differential
decoder. Moreover, the proposed decoder does not require any
knowledge of channel, signal, or noise power for decoding of
the space-time data, and it performs better than the differential
STBCs of [14], [15].

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1
k
sn ×

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k
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k
n n
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Receiver
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MIMO
Channel
Transmitter
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PARTIAL
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Fig. 1.Block diagram of the differential scheme with precoder.
2) We derive an upper bound of the pairwise error probability
(PEP) of differential coding for full-rank square OSTBC with
unitary and non-unitary constellations over arbitrarily corre-
lated Rayleigh channels. In addition, an approximate bound
of PEP of differential full-rank square non-orthogonal STBCs
is also derived.
3) We propose precoded differential coding to reduce the
effect of channel correlations by introducing a full memoryless
precoder matrix in the transmitter. This precoded differential
coding is applicable to full-rank square orthogonal and non-
orthogonal STBCs with unitary or non-unitary constellation
and arbitrarily correlated Rayleigh channels. Whereas, the
previously proposed precoder for differential codes [16] only
considers the special case of the Kronecker model with trans-
mit correlation only and full-rank square OSTBCs with M-
PSK constellations.
The rest of this paper is organized as follows: In Section II,
the system model is introduced. The non-unitary constel-
lations used by orthogonal and non-orthogonal STBCs are
discussed and the proposed encoding and decoding of the
differential STBC for non-unitary constellations are explained
in Section III. Section IV derives upper bound of PEP of
the differential OSTBCs and approximate bound of PEP
of the differential non-orthogonal STBCs with unitary and
non-unitary constellations over arbitrarily correlated Rayleigh
channels. The precoder is designed in Section V to improve the
performance of the system over correlated Rayleigh channels.
Simulation results are discusses in Section VI, and Section VII
contains some conclusions.
Notation: We have used the following notation throughout
the paper: Uppercase bold letters X are used for matrices,
lowercase bold letters x are used for vector, x, X denote
scalar variables, (·)Hmeans the Hermitian of a matrix, (·)Tis
used for the transpose of a matrix, E(·) represents expectation,
Pr{·} is used for probability, Tr{·} is the trace operator,
Frobenius norm of a matrix is depicted by ?·?2
identity matrix, |·| denotes the determinant operator, vec(·) is
vectorization operator, which stacks the columns of matrix in
a long column vector, rank(·) denotes the rank of the matrix,
max{.,.} choses larger out of the two quantities, and Re{·}
is used as the real operator, which returns only the real part
of the matrix it is applied to.
F, I stands for
II. SYSTEM MODEL
Consider a MIMO system with nttransmit and nrreceive
antennas as shown in Fig.1. Let H be a nr×ntchannel matrix
and Skbe the nt×n data matrix transmitted at block k. The
received data nr×n matrix will be
Yk= HSk+ Qk,
(1)
where Qkis an nr×n matrix, containing additive white
complex-valued Gaussian noise (AWGN), whose elements
are i.i.d. Gaussian random variables with zero mean and
variance σ2. The channel H is assumed constant over the
transmission period of at least two consecutive code matrices
(Sk−1and Sk).
A. Model of Correlated Rayleigh Channels
We assume flat block-fading correlated Rayleigh fading
channel model [17]. Let the channel H have zero mean,
complex Gaussian circularly distribution with positive semi-
definite autocorrelation given by R = E?vec(H)vecH(H)?
related Rayleigh channels can be found by vec(H) =
R1/2vec(Hw), where R1/2is the unique positive semi-
definite matrix square root [18] of R and Hwis nr×ntmatrix
consisting of complex circular Gaussian distributed elements
with zero-mean and unit variance.
Kronecker Model: A special case of the model given above
is the Kronecker model, which can be represented as [17]
of size ntnr × ntnr. A channel realization of the cor-
R = RT
t⊗ Rr,
(2)
where Rr is the nr × nr receive correlation matrix and
Rt is the nt× nt transmit correlation matrix. This model
does not always render the multi-path structure correctly, and
it might introduce artifact paths that are not present in the
underlying measurement data [19]. The Kronecker model can
underestimate the mutual information of the MIMO channel
systematically. However, the generalized model that we used
throughout this work considers that the receive (or transmit)
correlation depends on at which transmit (or receive) an-
tenna the measurements are performed. Hence, the generalized
model depicts the behavior of MIMO channel more accurately
than the Kronecker model. The previously proposed differ-
ential code based on eigen-beamforming precoder [16] only
considers the Kronecker model with transmitter correlation
and no receiver correlation and is not applicable for the general
correlation model assumed here.

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−2−1.5−1 −0.50 0.511.52
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re {⋅}
Im {⋅}
(a) Optimized non-unitary constellation of rate-2 STBC of [7].
−2−1.5 −1 −0.50 0.51 1.52
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
Re {⋅}
Im {⋅}
(b) Optimized non-unitary constellation of 8-QAM STBC of [9].
Fig. 2.Optimized non-unitary constellations of non-orthogonal STBCs.
III. DIFFERENTIAL STBC FOR NON-UNITARY
CONSTELLATIONS
A. Standard Non-Unitary Constellations Used by OSTBCs
Let sibe a symbol drawn from a non-unitary constellation
ψ like QAM, and let s1,s2,...,sns, ns≤ nt, be the block of
nssymbols transmitted through the OSTBC Ck[1], [3], [20],
then
CkCH
?
where a = 1, if Ck = GT
(7.4.8), (7.4.9), (7.4.10), (7.4.11)]. As CkCH
equal to Ck−1CH
differential STBC systems in [4], [5] cannot be used. However,
the QAM constellations provide higher data-rates and perform
better than the same rate PSK constellations [20], therefore,
they are more useful for improving the performance of the
communication system.
k= a
ns
i=1
|si|2Int,
(3)
2of [3] or OSTBCs of [20, Eq.
kis not always
k−1, Ckis a non-unitary code matrix and the
B. Optimized Non-Unitary Constellations Used by Non-
Orthogonal High Rate STBCs
Some examples of full-rank square non-orthogonal high-
rate codes are full-diversity, high-rate STBCs from division
algebras [7], STBC based on number theory [9], the Golden
codes [10], and the perfect STBCs [11]. In these STBCs,
the entries in the code matrix Ckare linear combinations of
the elements of regular QAM or PSK constellations. These
higher order constellations are optimized to maximize the
coding gain. Therefore, the algebraic high-rate codes perform
better than the OSTBCs. Fig. 2 shows the constellations of
rate-2 code of [7] and 8-QAM STBC of [9], which uti-
lize 4-QAM and 8-QAM, respectively, as the basic signal
constellations to obtain the optimized constellations. These
codes not only improve the spectral efficiency but also achieve
the diversity-multiplexing trade-offs [21]. But these codes are
non-orthogonal and the previously reported differential coding
schemes [4]–[6], [12]–[15] cannot be applied for them.
C. Differential Encoding for Non-Unitary Constellations
Let us first, consider a case, when channel is uncorrelated,
i.e., R = Intnrand no precoding F = Intin Fig. 1. Let
Vk−1be an nt×ntnon-orthogonal STBC matrix transmitted
at block k − 1. The received data matrix at block k − 1 will
be
Yk−1= HVk−1+ Qk−1.
(4)
In the next block k, we transmit the Vkmatrix as shown in
Fig. 1
1
βk−1Vk−1Ck,
Vk=
(5)
where Ckis a full-rank square non-orthogonal STBC matrix
of size nt× nt, which consists of data from the optimized
non-unitary constellation (see Subsection III-B), βk−1 is a
normalization factor, which limits the peak power of Vk.
The choice of βk−1 depends upon the specific code. Let
s1,s2,...,snsbe a block of nsoptimized non-unitary symbols
transmitted at block k−1 through differential non-orthogonal
STBC Vk−1, then one choice of βk−1could be
βk−1=
?
a
ns
?
i=1
|si|2
?1/2
,
(6)
where a is a constant, which depends upon the non-orthogonal
STBC and number of transmit antennas. a = 1/2 for rate-2
STBCs of [7, Ex. (6)] and STBCs of [9, Eq. (5)] for two
transmit antennas and QPSK constellation, and a = 1/5 for
Golden codes [10, Eq. (10)] with any-QAM used for obtaining
the optimized constellation and two transmit antennas. Another
choice of βk−1could be a constant, to put an average power
constraint. The received data at block k ≥ 1 will be
Yk= HVk+ Qk= HVk−1˜Ck+ Qk,
(7)
where˜Ck= Ck/βk−1. Here, it is assumed that the channel
H remains stationary for at least two consecutive code matrix
transmission. In general, we transmit the following data matrix
at any block p ≥ 1
Vp= Vp−1˜Cp= Vp−2˜Cp−1˜Cp= V0˜C1˜C2˜C3···˜Cp, (8)
therefore, this is a normal form of differential encoding
(similar to that of unitary constellations) where the current
transmitted code matrix has information about the previously
transmitted code matrices.

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D. Decoding of Non-Orthogonal Differential Space-Time
Code with Non-Unitary Constellations
It can be verified from (4) and (7), that if channel H and
Vkare known,
f (vec(Yk)|HVk) =
1
πnrntdet(Ψ)exp(−[vec(Yk)
−vec(HVk)]HΨ−1[vec(Yk)−vec(HVk)]
?
,
(9)
where Ψ = E?vec(Qk)vecH(Qk)?is the covariance matrix
?vecT(Yk),vecT(Yk−1)?T
and Ckare known,
of Qk. Since vec(Qk) is AWGN, Ψ = σ2Inrnt. Let y =
be a vector consisting of two
consecutive blocks at the receiver. When H, Vk−1, βk−1,
f (y|H,Vk−1,βk−1,Ck) =
1
π2nrnt(σ2)2nrnt
× exp
?
−1
σ2
k
?
l=k−1
?vec(Yl) − vec(HVl)?2
F
?
.
(10)
In order to find a maximum-likelihood (ML) estimate of the
unknown data Ck, we need to maximize (10). If we assume
that σ2is known, then the following metric is first minimized
with respect to (w.r.t.) all unknown quantities H, Vk−1, βk−1,
and, subsequently, over Ck:
Γ =
k
?
l=k−1
?vec(Yl) − vec(HVl)?2
F, k ≥ 1,
(11)
which is equivalent to maximization of (10). From the property
of the Frobenius norm [22], (11) can be alternately expressed
as
Γ =
???Yk− HVk−1˜Ck
maximization of joint p.d.f. of vec(Yk) and vec(Yk−1) and
it consists of two different norms. Whereas, the differential
codes of [12]–[15] use a least square decoder which assumes
Yk−1 as an estimate of HVk−1 and finds the estimate
of the OSTBC data by minimizing the following metric:
???Yk− Yk−1˜Ck
its performance is very poor as compared to the optimal
decoder. Therefore, we will proceed our analysis with the ML
metric obtained in (12). Assuming that˜Ckis known, the ML
metric in (12) can be minimized w.r.t. HVk−1by taking the
derivative of (12) w.r.t. HVk−1[23] and equating it to zero
as
???
2
F+ ?Yk−1− HVk−1?2
F.
(12)
It can be seen from (12) that the ML metric is obtained from
???
2
F. However, this least square decoder does
not work well in the case of non-orthogonal STBCs and
HVk−1=
?
Yk˜C
H
k+ Yk−1
??˜Ck˜C
H
k+ I
?−1
.
(13)
After substituting the value of HVk−1 from (13) into (12)
we get
Γ
?˜Ck
+
?
=
????Yk−
?
Yk˜C
H
k+ Yk−1
??˜Ck˜C
H
k+ I
?−1˜Ck
2
????
2
F
????Yk−1−
?
Yk˜C
H
k+ Yk−1
??˜Ck˜C
H
k+ I
?−1????
F
. (14)
In order to find ML estimate of Ck, (14) can be jointly
minimized w.r.t. βk−1 and Ck. The physical interpretation
of (14) is as follows: At a given time the receiver has informa-
tion about two consecutively received data matrices, i.e., Yk
and Yk−1. Hence, in contrast to the conventional differential
decoder it tries to minimize the euclidean distance between
Yk and HVk−1˜Ck, and Yk−1 and HVk−1, see (12). As
HVk−1 is not known, it first finds an estimate of HVk−1
through (13) in terms of˜Ck. It can be seen from (13) that the
estimate of HVk−1depends upon˜Ck, therefore, from (14)
the receiver tries to utilize the available information about Yk
and Yk−1in a good way as possible. It will be shown later
in (17) and (18) that for OSTBC, the ML decoder reduces into
a single norm which minimizes the euclidean distance between
Ykand Yk−1˜Ck, and Ykand Yk−1Ckfor non-unitary and
unitary constellations, respectively. If the receiver knows βk−1
perfectly, (14) can be minimized w.r.t. Ckto find ML estimate
of Ck. Therefore, the ML detection of Ckleads to
ˆCk= arg min
Ck∈ΞΓ
?˜Ck
?
,
(15)
where Ξ is the set of all full-rank square STBC matrices,
consisting of symbols belonging to the finite non-unitary
constellation.
Let si ∈ M-PSK constellation, then by setting |si|2=
1/(ans) it can be seen from (6) that βk−1 = 1, and ML
decoder of (15) reduces into
ˆCk=arg min
Ck∈A
?????Yk−
YkCH
?
YkCH
k+Yk−1
??
CkCH
k+I
?−1
Ck
????
2
F
+
????Yk−1−
where A is the set of all full-rank square STBC matrices,
consisting of symbols belonging to M-PSK constellation.
If Ck is a full-rank square OSTBC with M-QAM or M-
PAM constellation, it can be shown after some manipulations
that (14) reduces into the following form:
?
k+Yk−1
??
CkCH
k+I
?−1????
2
F
?
,
(16)
Γ
?˜Ck
?
=
?
β2
k−1
β2
k+ β2
k−1
?2???Yk− Yk−1˜Ck
???
2
F.
(17)
If Ckis a full-rank square OSTBC with M-PSK constellation,
then βk= βk−1= 1 and (17) reduces into
Γ (Ck) =1
4?Yk− Yk−1Ck?2
F,
(18)
which is an optimal decoder of Ckwith unitary constellations,
already obtained in the literature [20, Eq. (9.6.18)]. It is clear
from (14), (15), (16), (17), and (18), that the proposed decoder
is applicable to unitary, and standard and optimized non-
unitary constellations, and orthogonal and non-orthogonal full-
rank square STBCs. Hence, our differential decoder is more
general than the previously suggested differential decoder
for full-rank square OSTBCs with unitary or non-unitary
constellations. In addition, the proposed decoder does not need
any information about the channel power, signal power, or
noise power for detection of Ck.

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E. Application of the Proposed ML Decoder to Differential
Quasi-OSTBC
Quasi-OSTBC (QOSTBC) has been proposed in [24]–
[28]. The QOSTBC is a full-rank and square non-orthogonal
STBC with pairwise decoding complexity, i.e., symbols can
be decoded in pair. Differential schemes for QOSTBC have
been reported in [29], [30]. Recently, an optimized differential
modulation scheme based on the QOSTBC of [27], [28]
was proposed in [31]. In [31], an optimized constellation is
designed such that the differential encoding and suboptimal
decoding1proposed in [12] for unitary orthogonal matrices
can be applied to QOSTBC and full diversity, and rate-1
can be achieved with maximized coding gain. However, as
we have derived ML decoder for differentially encoded non-
orthogonal STBCs, therefore, it is now possible to apply the
simple encoding of (5) to the QOSTBC and the receiver can
use the ML decoder of (14). We will show the comparison of
the performance of the proposed differential encoding and ML
decoding for QOSTBC of [25] with the same rate optimized
differential QOSTBC scheme of [31] in Subsection VI-B. In
addition, the proposed differential scheme is more general than
the differential schemes in [29]–[31], since we also consider
the effect of the channel correlation on the performance of
the differential system and propose a precoder design for
improving the system performance.
IV. PEP PERFORMANCE ANALYSIS OF THE PRECODED
DIFFERENTIAL STBC
In this section, we consider the effect of channel correlation
on the system performance. So far, the effect of correlated
channels for differential STBCs has not been studied in
detail [16]. We present theoretical analysis, which considers
the effect of channel correlation on the PEP performance of the
differential communication system. The idea is as follows: To
improve the performance, we can apply precoding before the
differential signals are transmitted as done in the coherent non-
differential case [32], [33]. The block diagram of the precoded
differential coding scheme is shown in Fig. 1. Here, F is
the nt× n precoder matrix multiplied with the differential
code matrix before transmission. The precoder matrix does
not change the ML decoding as the precoding matrix can be
absorbed into the channel as H′= HF. Hence, the ML
decoding of Ckin a precoded system can still be done by (15)
and (16) without the knowledge of the precoder in the receiver.
A. PEP Bound for Precoded Differential STBC
We are using a precoder before transmission of the differen-
tial STBC as shown in Fig. 1, to reduce the effect of channel
correlation and improve the performance of the system. Hence,
the nr× n received data matrices at the (k − 1)-th and the
k-th block, corresponding to the precoded differential STBC
are
Yk−1= HFVk−1+ Qk−1,
(19)
1The suboptimal decoding here means that the receiver considers the
previously received data sample as noisy estimate of the channel and finds a
least square estimate of the currently transmitted data.
Yk= HFVk+ Qk= HFVk−1
Ck
βk−1
+ Qk,
(20)
where F is a memoryless precoding matrix of size nt×n, Ck
of size n × n is the full-rank square non-orthogonal STBC
matrix, and Vk−1 and Vk are differential code matrices of
size n × n. We may write (19) and (20) as follows:
˜Yk= [Yk,Yk−1]nr×2n= HFVk−1
?Ck
βk−1,In
?
?
n×2n
+?Qk,Qk−1
nr×2n,
(21)
where [ · ]l×mindicates the size of the matrix inside the
parenthesis as l × m. We can write (21) alternately as
˜Yk= [Yk,Yk−1]nr×2n= HF˘Vk−1[Ck,βk−1In]n×2n
+?Qk,Qk−1
?
nr×2n,
(22)
where˘Vk−1=˘Vk−2Ck−1has size n×n and is defined as:
˘Vk−1?Vk−1
βk−1
=V0
β0
C1
β1
C2
β2
···Ck−2
βk−2
Ck−1
βk−1.
(23)
The differential detection may be treated as a problem of
detecting Φk= [Ck βk−1In]n×2nwith an unknown channel
HF˘Vk−1. It can be proved with the help of the results given
in [34, Section III] that
EH
?Pr?C0
k→ Ck
??≤
ΦT
k
?Φ0
?C0
Φ∗
????Intnr+
1
4σ2
?
F∗˘V
∗
k−1
?Φ0
−1
,
k
?∗
(24)
×Π⊥
k
?T˘V
T
k−1FT⊗ Inr
?
R
???
= I2n −
where Φ0
k
=
?−1
k,βk−1In
?
n×2nand Π⊥
ΦT
k
ΦT
plement of the range space of ΦT
that the PEP of the differential non-orthogonal STBC depends
upon the product matrix˘Vk−1, which makes the PEP bound
of (24) impractical for precoder design. As˘Vk−1is a product
of k normalized matrices which depends upon k, therefore,
˘Vk−1 will be different at different block k and one need
to determine the value of˘Vk−1 which maximizes the PEP
for every block k which is computationally complex. If we
design a precoder based on this bound, then the precoder
matrix will depend upon˘Vk−1, i.e., in turn it depends upon
the block k. This means that the precoder matrix F will be
different for different block transmissions, hence, the effective
channel will be block varying and the proposed differential
scheme will fail. Therefore, for designing the precoder we
need to form the optimization problem such that it does not
depend upon˜Vk−1. Further, it can be seen from (24) that
the term
?Φ0
?
constellation or a fixed constant to satisfy the average power
constraint. Therefore, fixed values of Ck, C0
be obtained which maximize PEP irrespective of k.
In order to obtain a more practical bound of PEP we should
average (24) over˘Vk−1. Applying Minkowski’s determinant
k
?
Φ∗
kΦT
k
kis the orthogonal projector onto the com-
k. It can be seen from (24)
k
?∗Π⊥
Φk
?Φ0
k
?T
depends upon STBC matrices
Ckand C0
a?ns
k, and βk−1, which can be a vector norm βk−1=
i=1|si|2containing elements from a fixed finite QAM
k, and βk−1can

Page 6

6
inequality [35, Part II, Eq. (4.1.8.6)] to (24), we can further
upper bound the PEP of differential STBC as
EH,˘ Vk−1
?Pr?C0
?∗Π⊥
k→ Ck
??≤ E˘ Vk−1
k−1FT⊗Inr
??
?
1+
1
4σ2
???
?
F∗˘V
∗
k−1
×?Φ0
By using the property of the determinant |AB| = |BA| =
|A||B| = |B||A|, (25) can be written as
k
ΦT
k
?Φ0
k
?T˘V
T
R
???
1
ntnr?−ntnr?
, (25)
EH,˘ Vk−1
?Pr?C0
F∗?Φ0
k→ Ck
??≤E˘ Vk−1
?TFT⊗Inr
??
1+1
4σ2
???˘V
T
k−1˘V
∗
k−1
???
1
ntnr
×
???
?
k
?∗Π⊥
ΦT
k
?Φ0
k
?
R
???
1
ntnr?−ntnr?
. (26)
In the case of OSTBC utilizing non-unitary constellation, it
can be shown by using the unitary property of normalized
OSTBC that˘Vk−1˘V
upper bound of PEP (UBPEP) for differential OSTBC with
non-unitary constellation can be given as
∗
k−1= In. Therefore, from (26) the
UBPEPO=
?
1 +
1
4σ2
???
?
F∗?Φ0
k
?∗Π⊥
⊗Inr)R|
ΦT
k
?Φ0
1
ntnr
k
?TFT
?−ntnr, (27)
which is free from˘Vk−1. By using the inequality 1+a ≥ a,
we can obtain UBPEP of non-orthogonal STBC from (26) as
UBPEPNO= E˘ Vk−1
????˘V
T
k−1˘V
∗
k−1
???
−1????
?
F∗?Φ0
?−ntnr
k
?∗Π⊥
.
ΦT
k
×?Φ0
k
?TFT⊗ Inr
?
R
???
−1?
1
4σ2
(28)
It can be seen from (28) that E˘ Vk−1
is a constant which does not depend upon F, there-
fore, the PEP can be minimized with respect to (w.r.t.)
F
by only maximizing the following term w.r.t. F:
???
of OSTBC also PEP can be minimized by maximizing
???
it is difficult to find E˘ Vk−1
????˘V
T
k−1˘V
∗
k−1
???
−1?
?
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
?
R
???, which is free from
???.
???
k−1
˘Vk−1. Similarly, it can be seen from (27) that in the case
?
F∗?Φ0
In the case of non-orthogonal STBC,˘V
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
????˘V
−1
. However, by applying Jensen’s in-
?
R
T
k−1˘V
−1?
????˘V
∗
k−1?= Inand
. Therefore, we
T
k−1˘V
∗
k−1
may take an approximation that E˘ Vk−1
T
k−1˘V
∗
???
−1?
≈
???E˘ Vk−1
this approximation lowers the upper bound and it can result
into a lower bound of PEP as well. For normalized non-
orthogonal STBCs [7], [24], [25], E˘ Vk−1
and an approximate bound of PEP (ABPEP) can be obtained
?˘V
T
k−1˘V
∗
k−1
????
equality [36, Eq. (2.76)], it can be seen from (28) that
?˘V
T
k−1˘V
∗
k−1
?
= In
from (28) as
ABPEPNO=
?
1
4σ2
?−ntnr ???
?
F∗?Φ0
k
?∗Π⊥
⊗Inr)R|−1, (29)
ΦT
k
?Φ0
k
?TFT
which does not depend upon˘Vk−1.
Let us define Θ =
value of Θ which maximizes the PEP with the trivial pre-
coder F =
?
correlation matrix. Here, [Imax{nt,n}]nt×n is a matrix of the
size nt× n, taking the first nt rows and first n columns
from an identity matrix Imax{nt,n}, where max{nt,n} returns
the maximum value of ntand n. Therefore, from (27), (28),
and (29) Θ can be obtained as
?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?T
and choose the
P
ansσ2
s[Imax{nt,n}]nt×n and invertible channel
Θ =
min
∗Π⊥
¯ Θ=(¯Φ0
k)
¯ ΦT
k(¯Φ0
k)
T
¯ C0
k?=¯ Ck
???¯Θ ⊗ Inr
?R??,
(30)
where¯Φ0
k=
?¯C0
k
¯βk−1In
?
n×2n,¯Φk=?¯Ck¯βk−1In
?
n×2n,
¯C0
either a vector norm of nssymbols belonging to the unitary
or non-unitary constellation, or a fixed constant.
kand¯Ck are two different STBC matrices, and¯βk−1 is
V. DESIGN OF PRECODER FOR DIFFERENTIAL STBCS
In this section, we will discuss the design of the precoder
for differential STBCs. Since the channel statistics varies
far more slowly than the channel coefficients, we assume
that the receiver can estimate the channel correlation matrix
and noise variance, and feed these back to the transmitter.
Under the assumption that the transmitter knows the channel
correlations and the noise variance perfectly, we will discuss
the precoder design for the differential STBCs using unitary
or non-unitary constellations and full-rank square orthogonal
or non-orthogonal STBC.
A. Problem Formulation
By using the properties of full-rank square orthogonal and
non-orthogonal STBC [7]–[11], it can be proved that the
average power transmitted per orthogonal or non-orthogonal
STBC block is
E
?
CkCH
k
?
= ansσ2
sIn,
(31)
where σ2
data vector s to be encoded into non-orthogonal STBC. The
average power constraint on the transmitted block Sk= FVk
can be expressed as
sis the average power of each symbol in the ns× 1
ansσ2
sTr
?
FFH?
= P,
(32)
where P is the average power used by the transmitted
block Sk, The goal is to design a precoder F such that
an upper bound of the pairwise error probability (UBPEP)
is minimized under the constraint over the average power
on the transmitted block Sk given in (32). It can be seen
from (27) and (28) that UBPEP can be minimized by

Page 7

7
maximizing
ever, from the property of the determinant |AB|
|BA|=|A||B|=|B||A| it can be shown that
the trivial precoder F
=
a solution of the optimization problem of maximizing
???
coding solution for differential STBCs, we need to modify
the optimization problem. By using the determinant inequality
|I + A| ≥ |A| [20, Eq. (A.4.26)], it can be shown that
???
?
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
?
R
???. How-
=
?
P
ansσ2
s[Imax{nt,n}]nt×n is
?
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
?
R
??? under the power
constraint of (32). Therefore, for obtaining a non-trivial pre-
???
?
≤
F∗?Φ0
???Intnr+
side of (33) for designing the precoder for differential STBC.
Apperantly, utilizing the upper bound of (33) for minmizing
the PEP may result into a suboptimal precoding solution
and obtaining an optimal precoder is still an open research
problem. However, it is shown by the simulation results in
Subsections VI-D and VI-E that the proposed precoder is
able to provide significant performance improvement to the
differential STBCs over correlated MIMO channels and better
results are obtained with the precoder found from the right
hand side of (33) as compared to the trivial precoder. We can
express the optimization problem as [32], [33], [37], [38]
k
?∗Π⊥
?
ΦT
k
?Φ0
k
k
?TFT⊗ Inr
?∗Π⊥
?
R
???
F∗?Φ0
ΦT
k
?Φ0
k
?TFT⊗ Inr
?
R
???. (33)
We propose to maximize the upperbound given in the right
max
{F∈Cnt×n}ln
???Intnr+
?
F∗?Φ0
=
k
?∗Π⊥
P
ansσ2
ΦT
k
?Φ0
k
?TFT⊗Inr
?
R
???,
subject to Tr
?
FFH?
s
.
(34)
It can be observed that the use of natural logarithm does not
change the nature of the objective function because ln(·) is a
monotonically increasing function. It can be seen from (34),
that a closed-form precoder is difficult for the general case of
arbitrary correlation. Hence, we focus on fast-converging nu-
merical method to design the precoder in the next subsection.
B. Precoder Design
The constrained minimization problem of Subsection V-A
can be converted into an unconstrained minimization problem
by introducing a Lagrange multiplier µ′:
L(F) = ln
???Intnr+
?
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
+ µ′Tr
k
?TFT⊗Inr
?
?
. (35)
R
???
FFH?
Since the objective function should be maximized, µ′< 0. It
can be shown after some manipulations that [23] the necessary
conditions for optimality can be written as:
vecT(F) = µDF∗ln|Intnr
?
where DF∗ is the first order derivative with respect to F∗[23],
and µ is a scalar chosen such that the power constraint in (32)
is satisfied. The derivative can be found with the help of results
+
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
?
R
???,
(36)
TABLE I
PSEUDO-CODE OF THE GENERAL NUMERICAL OPTIMIZATION
ALGORITHM.
Step 1:
Initialization
Choose space-time code, values for P, a, n,
σ2
Estimate the channel correlation matrix R
and noise variance σ2
Initialize the precoder F to an already
optimized precoder or to
?
Precoder Optimization
i := 0
repeat
i := i + 1
Calculate the right hand side of (37).
Normalize the precoder matrix Fi
such that: ?Fi?F=
to satisfy the power constraint.
until
?Fi− Fi−1?F< ǫ
The optimized precoder is given by F = Fi
s, nt, nr, ǫ, and the signal constellation
F0=
P
ansσ2
s[Imax{nt,n}]nt×n
Step 2:
?
P
ansσ2
s,
in [23]. We can summarize the results of the derivative as
follows [23]:
DF∗ln
???Intnr+
?
F∗?Φ0
k
?∗Π⊥
ΦT
k
?Φ0
k
?TFT⊗ Inr
RHSH?
?TFT⊗ Inr
?TFT⊗Inr
?
R
???
= vecH?
?Φ0
?Φ0
Π,
(37)
where R =
(In⊗ Knr,nt⊗ Inr)[Innt⊗ vec(Inr)],
?
and Knr,nt is the commutation matrix of size nrnt ×
nrnt[22].
A description of the procedure of minimizing the upper
bound of the PEP with respect to the precoder F is sum-
marized with pseudo-code in Table I.
??Φ0
??Φ0
k
?∗Π⊥
?∗Π⊥
ΦT
k
k
?
R, Π
=
S =
Intnr+
k
ΦT
k
k
?
R(F∗⊗Inr)
?−1
,
VI. SIMULATION RESULTS
The simulations are performed with 4, 8, 16, and 64
standard QAM and optimized non-unitary constellations, PSK
constellations, nt∈ {2,3,4}, and nr ∈ {1,2}. The channel
is assumed to be circular symmetric complex Gaussian with
zero mean, and with uncorrelated coefficients except in Sub-
sections VI-D and VI-E, where we consider the channel to be
correlated. The channel is also assumed stationary over the
transmission period of two consecutive code matrices. The
initialization matrix is kept as identity, i.e., V0 = I2 and
assumed to be perfectly known to the receiver. The SERs
at each SNR level are obtained through 106Monte-Carlo
simulation runs.

Page 8

8
5 1015 20 2530 35
10
−4
10
−3
10
−2
10
−1
10
0
SNR [dB]
SER


64−PSK
64−QAM
16−PSK
16−QAM
8−QAM
8−PSK
Fig. 3.
orthogonal designs.
Performance of the proposed differential space-time code for
A. Performance of the Proposed Differential Scheme with
OSTBCs
Fig. 3 shows the performance of the proposed differential
STBC with OSTBC (Alamouti code), nt = 2, and nr = 1.
The symbols are chosen from 8, 16, and 64 standard QAM
and PSK constellations. Fig.3 shows that our differential code
works well with OSTBCs. The performance of the proposed
differential code is also compared with unitary differential
code [4], [5] with similar rate PSK constellations. The pro-
posed code works better than unitary differential codes in the
case of higher order QAM like 16 and 64-QAM. However,
there is no performance gain in the case of 8-QAM compared
to the 8 PSK as the two lower curves (8-QAM and 8-PSK) in
Fig. 3 are almost on the top of each other.
B. Performance of the Proposed Differential Scheme with
Non-Orthogonal STBCs
Fig. 4 shows the performance of the proposed differential
scheme with nt= 2, nr= 1, different high-rate STBCs [7],
[9], [10], and the ML decoder of (15). The simulations are
performed with rate-2 STBCs from field extension [7], 4-QAM
and 8-QAM STBCs based on number theory [9], and 4-QAM
Golden codes of [10]. The codes of [7] use the symbols from
higher order non-unitary optimized 16-point constellation,
extended over QPSK. Similarly, the 4-QAM and 8-QAM
STBCs of [9] use 16-point optimized constellation obtained
from 4-QAM and 64-point optimized constellation obtained
from 8-QAM, respectively, and 4-QAM Golden codes [10]
use optimized 16-point constellation obtained from 4-QAM.
The performance of the proposed differential scheme for these
codes is compared with the same rate PSK constellations,
where in place of the optimized constellation, the high-rate
STBCs [9] use the symbols drawn from PSK constellation.
From Fig.4, it is seen that differential coding for non-unitary
constellations performs better than the differential coding for
unitary constellations using the same rates. Moreover, the pro-
posed differential scheme works better in the case of Golden
5 101520 25 30
10
−3
10
−2
10
−1
10
0
SNR [dB]
SER


64−PSK
8−QAM STBC of [9]
16−PSK
Rate−2 STBC of [7]
4−QAM STBC of [9]
Suboptimal decoding of 4−QAM Golden code of [10]
4−QAM Golden code of [10]
Fig. 4.
orthogonal STBCs.
10
Performance of the proposed differential space-time code for non-
05 10 152025
10
−4
10
−3
10
−2
10
−1
0
SNR [dB]
SER
Differential STBC of [39]
Proposed Differential coding for STBC of [7]
Fig. 5.
and Differential STBC of [39] with three transmit and one receive antennas.
Comparison of the proposed differential coding with STBC of [7]
code than the other high data-rate codes. In addition, our
differential code is working well for unitary and non-unitary
constellations. We have also plotted the SER performance of
a suboptimal decoder for 4-QAM Golden codes in Fig. 4. The
suboptimal decoder assumes that the previously transmitted
data is a noisy estimate of the channel and tries to minimize
the following metric to find the estimate of current data:
???Yk− Yk−1˜Ck
the ML decoder for Golden code and at high SNR it looses
diversity.
In Fig.5, we have compared the performance of the differ-
ential code of [39] with the proposed differential scheme with
STBC of [7] for three transmit and one receive antennas case.
The differential code of [39] uses the unitary matrices given
in [39, Subsection V-C] for three transmit antennas with data-
rate of 1.9924 bits/sec/Hz. Whereas the STBC of [7] utilizes
the full rank code for three transmit antennas given by [7,
???
2
F. From Fig. 4, it can be seen that the
suboptimal decoder does not perform well as compared to

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9
05 10152025
10
−5
10
−4
10
−3
10
−2
10
−1
SNR [dB]
SER
Differential QOSTBC of [31]
Proposed differential scheme for QOSTBC
Fig. 6.
four transmit and one receive antennas.
Performance of the proposed differential scheme for QOSTBC with
Eq. (6)] with the indeterminate2z = e1.05and data-rate of
2 bits/sec/Hz. From Fig. 5, it can be seen that the STBC
of [7] with the proposed differential encoding and decoding
not only provides better data-rate but also performs better than
the differential code of [39] for all values of SNR.
The performance plot of the proposed differential scheme
for the QOSTBC of [25] with four transmit and one receive
antennas is shown in Fig. 6. The symbols are chosen from two
different QPSK constellations which have a relative rotational
angle of φ between them. The value of φ is kept as 0.5344 as
suggested in [25, Fig. 1]. We have also plotted the performance
of the same rate optimized differential QOSTBC of [31] which
utilizes the symbols chosen from the four point optimized
constellation shown in [31, Fig. 4 (a)]. From Fig. 6 it can
be seen that the proposed differential coding works approx-
imately 2 dB better than the differential QOSTBC of [31] at
SER=10−4.
C. Comparison with Conventional Differential Codes for
QAM Constellations
Fig. 7 shows the comparison of the proposed differential
code with the differential codes of [14] and [15]. The sim-
ulations are done for 16-QAM constellation, the Alamouti
code, nt = 2, and nr = 1. The decoding of differential
code of [14] depends upon the knowledge of the channel
power. The method of channel power estimation, suggested
in [14], requires the channel to be constant for more than
100 symbol durations (L ≥100), for obtaining good estimate
of channel power. If the channel remains constant for less
than 100 symbol durations, then the performance of [14],
degrades substantially as shown in Fig. 7. We have shown
the performance of [14] with L=2 and L=6. Since our code
is independent of channel power knowledge, it works equally
well with any length of channel if it is constant over at least
two consecutive block channel uses. The code in [15] uses
a soft decoder with a complex two level decoding structure.
2For more details see [7].
0510 15 202530
10
−4
10
−3
10
−2
10
−1
10
0
SNR [dB]
SER


Unitary Diff. STBC with 16−QPSK
Diff. STBC of [14] for 16−QAM with L=2
Diff. STBC of [14] for 16−QAM with L=6
Diff. STBC of [15] for 16−QAM
Proposed Diff. STBC for 16−QAM
Coherent detection of 16−QAM
Fig. 7. Comparison of the proposed differential STBC with the conventional
differential codes with QAM constellations.
0510 15 20 2530
10
−3
10
−2
10
−1
SNR [dB]
SER
Sim. SER without precoder
Sim. SER with Eigen−Beamforming [16]
Sim. SER with proposed precoder
Fig. 8.
unitary constellation with transmit correlations.
Performance of the precoded differential code for OSTBCs with
Nevertheless, the performance of this code is 3.7 dB poorer
than the coherent detection, even when the receiver perfectly
knows the signal power and the noise power. The proposed
code performs better than the differential STBC of [15] as
well.
D. Performance of Precoded Differential OSTBC with Unitary
Constellations
For making a comparison with the previously proposed
precoded differential system [16] we have plotted the SER
plots considering Kronecker correlation model with transmit
correlation only, i.e., [Rt]i,j= (0.99999)i−j,
ntand Rt= Inr, nt= 2, nr= 1, and QPSK constellation in
Fig. 8. It can be seen from Fig. 8 that the differential system
with our precoder works similar to the differential system with
the eigen-beamforming precoder [16]. Fig. 9 shows the per-
formance of precoded differential code for OSTBC (Alamouti
Code) with unitary constellation in a correlated non-Kronecker
1 ≤ {i,j} ≤

Page 10

10
0510 1520 2530
10
−3
10
−2
10
−1
SNR [dB]
SER
Sim. SER without precoder
Sim. SER with proposed precoder
Fig. 9.
unitary constellation in arbitrarily correlated channels.
Performance of the precoded differential code for OSTBCs with
channel with [R]i,j= (0.99999)i−j,
We have plotted the results for the QPSK constellation and a
MIMO system with nt= 2 and nr= 2. It can be seen from
Fig. 9 that the application of the proposed precoder improves
the SER versus SNR performance of the differential system for
correlated non-Kronecker channels by approximately 2.5 dB.
However, the precoder based on the eigen-beamforming [16]
is not applicable for arbitrarily correlated channels as it is
designed by making the assumption of a Kronecker model with
transmit channel correlations only. Our precoded differential
system works for any correlation model for Rayleigh channel
that has an invertible correlation matrix.
1 ≤ {i,j} ≤ ntnr.
E. SER Performance of Precoded Differential STBC with Non-
Unitary Constellations
Fig. 10 shows the symbol error rate (SER) performance
of the precoded differential space-time scheme for Golden
code and Alamouti code with nt = 2 and nr = 1. We
have performed simulations for a correlated channel using
[Rt]i,j= (0.99999)i−j,1 ≤ {i,j} ≤ nt and Rr = Inr.
The comparisons are made against a system not employing
any precoding, i.e., F =
?
is found by the procedure explained in Subsection V-B. It can
be seen from Fig. 10 that the proposed precoded differential
coding outperforms the differential coding without a precoder
for both orthogonal and non-orthogonal STBCs with non-
unitary constellations. The previously proposed precoder for
differential STBCs [16] is not applicable in this case, as it
only works with OSTBCs using unitary (PSK) constellations.
P
ansσ2
sInt. The precoding matrix
VII. CONCLUSIONS
We have proposed differential space-time block codes
that are applicable for full-rank square orthogonal and non-
orthogonal STBCs, and standard or optimized signal constel-
lations. The decoder of the proposed code does not need to ac-
quire any knowledge about the channel power, signal power, or
05 101520 25 30
10
−3
10
−2
10
−1
SNR [dB]
SER
2−QAM
Golden Code
8−QAM
OSTBC
Fig. 10.
non-orthogonal and orthogonal STBCs in correlated channels with non-unitary
constellations, nt= 2, and nr = 1: Unprecoded differential coding simulated
SER − ∗ −, precoded differential coding simulated SER − ◦ −.
Comparison of the precoded and unprecoded differential code for
noise power before the actual decision. Therefore, the decision
is based on two consecutive received data samples only, which
is an inherent property of a differential decoder. We have also
proposed a PEP based precoder design criterion for arbitrarily
arbitrarily invertible correlation matrix and orthogonal and
non-orthogonal full-rank square STBC with unitary and non-
unitary constellations, where existing precoder design cannot
be used.
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