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International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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STBC-SM BASED RELATIVE ESTIMATION OF
GOLDEN AND DSTTD CODES
R.P.Yadav GarimaMathur
V.C. R.T.U., KOTA Head, EC Deptt, JEC Kukas, Jaipur,
garimamathurjec@gmail.com
Sarita Boolchandani ShrutiSikarwar
M.Tech JEC Kukas, RTU, Kota, M.Tech student, JNU, Jaipur
saritaboolchandani@gmail.com sikarwar.shruti@gmail.com
ABSTRACT: The Golden code has recently been proposed as a 2x 2 space time block code that achieves the optimal diversi-
ty- multiplexing gain tradeoff for a multiple antenna system. Double Space Time Transmit Diversity (DSTTD) is an open loop
MIMO system with 4 transmits antennas. DSTTD achieves the best performance in rich scattering channels, spatial correlation
degrade the performance. Spatial modulation (SM) is a low complexity modulation scheme which is recently proposed for
multiple antenna wireless systems. SM is unable to achieve transmit diversity but it provides multiplexing gain by averting in-
ter channel interference with respect to single antenna system.
The aim of this paper is to present the design of multiple–antenna wireless systems that exploit the SM concept and can
achieve transmit–diversity gain. A new modulation concept STBC-SM is discussed which outperformed Golden and DSTTD
codes at higher SNR levels.
Keywords: Golden code, Double Space Time Transmit Diversity (DSTTD), spatial modulation (SM), Space Time Block
Code- Spatial Modulation (STBC-SM).
I. INTRODUCTION
A space time code method is used to enhance the
data transmission consistency in wireless commu-
nication systems through the use of multiple trans-
mit antennas. In this method, numerous surplus
copies of a data stream are transmitted to the re-
ceiver with the expectation that at any rate few of
them survive the physical path effects between the
transmission and reception in a condition that per-
mits reliable decoding.
Sethuramanet. al. [1] proposed a methodology for
designingfull-diversity high-rate LD codes using
cyclic division algebras.A division algebra is used
to provide a structured set ofinvertible matrices to
construct LD space-time codes. Usingthis tech-
nique, Belfioreet. al. [2] developed the Golden
Code, a2x 2LD code that provides both diversity
gain and full-rate.
In DSTTD scheme [3] signal on each transmit an-
tenna is affected by the interference from two out
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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of three otherantennas. Also, due to STTD encod-
ing, each data symbol is guaranteed to have
(transmit) diversity of 2.It is easy to see that the
regular STTD is a special case of DSTTD. At the
receiver, thesignal at each receive antenna is de-
spread. The signals from all receive antennas after
despreading arecoherently combined using two
STTD decoders for each receive antenna.
Mesleh introduced Spatial Modulation (SM) which
is a low complexity and advanced method. SM
uses an additional domain in the form of spatial
domain along with the time and space domain
along with the time and space domain for transmit-
ting information [4]-[11]. As a result, SM has
emerged as a transmission technology with higher
spectral efficiency along with an equivalent code
rate higher than one [12]. It exploits the multiplex-
ing gain offered by multiple transmit antennas, but
it does not exploit the MIMO system in terms of
transmit diversity potentials. This led to introduc-
ing Spatially Modulated Space Time Block
Codes(SM-STBC) which is designed such that it
takes advantage of both STBC and SM, and avoids
their disadvantages.
II. GOLDEN CODE
In most of the communication systems, size and
power present a constraint over employing more
than two antennas. Therefore, systems having two
transmitter antennas and two receiver antennas
prove to be of immense practical significance.
Likewise, the aspiration for soaring spectral effi-
ciency gives motivation for space-time codes hav-
ing high rate. Hence, a full-rate space-time code
having two-input two-output channels present a
class of codes which is of great significance and
thus, 802.16e standard [13] incorporates this class
of codes.
Earlier, the space-time codes designed were aimed
at either increasing the diversity gain or the multip-
lexing gain [14-16]. In terms of SNR, the golden
codes aimed at achieving an objective error proba-
bility, are found to perform better as compared to
all beforehand available full-rate codes consisting
of two transmit antennas.
System Model
We are considering space-time coding in which
transmission of four compound information sym-
bols is taking place for a period of two symbol pe-
riods for a system having two transmit antennas
and two receive antennas. The following 2x2 ma-
trix represents the transmitted code word:
(1)
Where ci[j] represents the symbol transmitted from
antenna i {1,2} at time j {1,2}. The signal re-
ceived at the receiver antenna k {1,2} at time j is
yk[j] and is given as:
(2)
Where wk[j] represents complex additive white
Gaussian noise at any time j at receiver antenna k
and hi,k is the channel coefficient between the ith-
transmit antenna and kth
receive antenna at time j.
For quasi-static fading, hi,k[j] is independent of
time j, i.e. hi,k[j] = hi,k.
For the two-input two-output channel, the highest
spatial diversity order is given by the product of the
number of transmitter antennas and number of re-
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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ceiver antennas, which in this case comes out to be
4. Two design criterions were obtained in [15] for
space-time codes over quasi-static fading channels,
which are as follows:
Rank Criterion: The difference matrix ∆C= C-
should have rank 2 for any match up of discrete
codewordsC and in order to attain fourth order
diversity.
Determinant Criterion: To optimize the perfor-
mance further, such a code should be chosen which
has full diversity so that it maximizes the coding
gain. The expression for coding gain for 2x2 square
matrices simplifies as:
(4)
Analogous criterions were proposed for rapid-
fading channels in [3].
III. DOUBLESPACE TIMETRANSMITDI-
VERSITY (DSTTD)
Double space time transmit diversity (DSTTD) is
an amalgam scheme which makes use of the prac-
tices of both spatial multiplexing and transmit di-
versity in one system [17]. A DSTTD system com-
prises of 2 two-antenna groups having a total of
four transmit antennas, where an orthogonal space
time transmit diversity (STTD) encoder has an as-
sociation with two antennas from each group.
Across the antenna groups, spatial multiplexing is
taken up, i.e. different groups send distinct data
streams. A very blunt trade-off is provided by the
DSTTD technique between transmit diversity and
spatial multiplexing.
System Model
In figure 1, block diagram of a DSTTD system is
shown with Nt = 4 of transmission antennas and Nr
≥ 2 of receiver antennas. Two data streams are then
obtained from the input information symbols after
demultiplexing. Further, each stream in encoded by
orthogonal STTD encoder.
Figure 1: Block diagram of DSTTD system
Output obtained at two successive periods t1 and t2
from two orthogonal STTD encoders can be embo-
died by a matrix of size 4X2 as given
(5)
In the equation, (·)∗implies the operation complex
conjugate, (·)T
symbolizes matrix transpose and xi
denotes an M-ary modulated symbol with Es as
power. In the matrix C, for each element being
symbolized by cij, column index j represents time
instant tj at j=1 or 2. Similarly, symbol in ith row
of C will be sent out by ith transmission antenna
where i ∈ {1, 2, 3, 4}.
The signal transmitted over the channel is distorted
by both additive noise and multi-path fading. In or-
der to keep the orthogonality of the STTD encod-
ing scheme intact, it is assumed for the channel to
be varying at such a slow pace that fading remains
steady for successive symbol periods [14].
Let us assume that fading coefficient between mth
transmission antenna and nth
receiver antenna is
represented by hnm, then at any time instant tj, the
signals received by the nth
receiver antenna may be
represented as:
for j= 1,2 (6)
where nj is the signal received at time instant tj by
nth
receiver antenna, znj is the equivalent additive
white Gaussian noise (AWGN) component with
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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variance N0, and cj represents the jth
column of the
encoded data matrix C. after simple algebraic
treatment of equation 1 and 2, the following equa-
tion is obtained
(7)
or in matrix format
for n= 1,2.. , Nr (8)
An input-output relationship of the system is di-
rected by stacking up the Nr receive vectors rn,
which follows as
(9)
In equation (4), a corresponding system is
represented with 4 transmitter antennas and 2Nr re-
ceiver antennas by a spatially multiplexed MIMO
system. Across the transmission antenna, 4 input
streams {x1, x2, x3, x4} are spatially multiplexed.
Equivalently, with each fading vector pertinent to
any of the 4 data streams, the comparable channel
matrix H is having four column fading vectors {h1,
h2, h3, h4}.
Based upon the 2 STTD encoders, the four trans-
mission streams can further be subdivided into 2
groups. The first group which is related with the
first STTD encoder consists of the first and second
data streams {x1, x2}, and the third and fourth data
streams {x3, x4} are in group 2 associated with the
second STTD encoder. The channel vectors which
belong to the same transmission group are ortho-
gonal with respect to each other due to the ortho-
gonality of the STTD encoder, i.e.
(10 a)
(10 b)
Nonetheless, data streams belonging to different
transmission groups still have interferences be-
tween them, and due to this interference the
DSTTD system performance will be solemnly in-
fluenced.
IV. SPATIALLYMODULATED SPACE
TIME BLOCK CODES (SM-STBC)
The code matrix is defined as:
(11)
where columns and rows correspond to the transmit
antennas and the symbol intervals, respectively.
Alamouti is chosen as the core STBC as it is ad-
vantageous in terms of spectral efficiency and sim-
plified ML detection. In this, two complex infor-
mation symbols (k1 and k2) from a -PSK or -
QAM constellation are transmitted using two
transmit antennas in two symbol intervals orthogo-
nally by the code word. For the SM-STBC scheme
the matrix in (11) is extended to the antenna do-
main with the help of an example.
Let us consider a MIMO system with four Nt
which transmit the Alamouti STBC from one of the
following four code words using BPSK modulation
scheme:
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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whereKabare the SM-STBC code word present in
the SM-STBC codebook , a = 1, 2 that do not
interfere with each other. The resulting SM-STBC
code is . ,c = 1, 2. . .z ,
b ≠ c is defined as non-interfering code word
group, that is they have no overlapping columns.
Maximum diversity and coding gain can be ob-
tained by optimization of the rotation angle, (12)
at the expense of expansion of the signal constella-
tion. If we do not consider , transmit diversity of
the order of 1 will be reduced because of overlap-
ping columns of codeword pairs. If this system is
generalized to M-ary signals, different codewords
are obtained which have M2 different realizations.
So, the spectral efficiency for four transmit anten-
nas of the SM-STBC scheme will be
= (1/2) log242 = 1 + log2 bits/s/Hz,
the factor 1/2 normalizes for the two channel uses
spanned by the matrices in (12). Because of this
normalization factor spectral efficiency of STBCs
using large no. of symbol intervals will be de-
graded as the no. of bits carried by antenna mod-
ulation (log2x), (where xis the total number of an-
tenna combinations) is normalized by the number
of channel uses.
SM-STBC System Design
A MIMO system with Nt transmit and Nr receive
antennas in the presence of a quasi-static Rayleigh
flat fading MIMO channel is considered. Minimum
coding gain distance (CGD) between two SM-
STBC codewords and , is defined as
(13)
where is transmitted and is erroneously de-
tected. The minimum CGD between two code-
books and is defined as
(14)
For SM-STBC code minimum CGD is given as
(15)
The minimum CGD between non-interfering co-
dewords of the same codebook is always greater
than or equal to the right hand side of (15).
The total number of codeword combinations in
SM-STBC should be an integer power of 2. To
provide design flexibility the pairwise combination
of transmit antenna is chosen from Nt available
transmit antenna so the number of transmit anten-
nas need not be an integer power of 2. Following is
an algorithm to design STBC-SM scheme:
Total number of codewords is calculated from the
given total number of transmit antennas, as
where p is a positive integer.
From each codebook ,a= 1, 2…n – 1 calculate
the number of codewords by using
and the total number of codebooks from
.The last codebook is calculated as ′ = x
- ( -1).
Codebook which contain anoninterfering code-
words is constructed as
(16)
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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for 2≤ a ≤ is constructed by considering the
following two important facts:
a) Out of Nt available antennas choose a codebook
containing non- interfering codewords.
b) Codewords used for construction of codebook
should not be used previously.
Rotation angle a that maximize min ( ) in (11)
for each , 2 ≤ a ≤ , is given as:
where
The spectral efficiency of the SM-STBC scheme is
given as:
[Bits/s/Hz] (17)
where x is the no. of antenna combinations.
In Figure 2, both STBC symbols and the indices of
the transmit antennas which carry information are
shown. In SM-STBC transmitter, 2m bits
enter during each two consecutive symbol intervals
where the antenna pair position is determined from
first log2c bits
while the last 2log2 bits determine
the symbol pair (k1, k2) ∈ 2. There is an increment
of ½ log2 x bits/s/Hz in spectral efficiency of SM-
STBC scheme when compared with that of Ala-
mouti scheme.
Figure 2: Block Diagram of SM-STBC transmitter
SM-STBC Decoder
Consider a MIMO system with Nt transmit and Nr
receive antennas in the presence of a quasi-static
Rayleigh flat fading MIMO channel to formulate
the ML decoder. From a given constellation M
such as, PSK or QAM that is assumed to have unit
energy, L symbols K1, K2… Klare chosen random-
ly and independently to form an input symbol se-
quence .The received 2xNr signal
matrix Y, can be expressed
as [18]
(18)
where is the 2 × Nt SM-STBC transmission
matrix for 2 channels, is a normalization factor
and is the average SNR at each receive antenna.
H and N denote the channel matrix and 2
× Nr noise matrixes, respectively. Let us assume H
and N are independent and identically distributed
(i.i.d.) complex Gaussian random variables with
zero means and unit variances. Also, assume that
receiver has perfect knowledge of H and remains
constant during the transmission of a codeword
[19]. SM-STBC code has xcodewords, from which
xM2different transmission matrices can be con-
structed using Nttransmit antennas. All possible
xM2transmission matrices are searched and ML
decoder decides in favor of the matrix that mini-
mizes the following metric:
(19)
Equation (19) is minimized due to the orthogonali-
ty of Alamouti‟s STBC as follows. The following
equivalent channel model is obtained by the decod-
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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er which extracts the information symbol vector
from Equation (12):
(20)
where is the 2Nr×2 equivalent channel matrix
[8] of the Alamouti coded SM scheme, which has
different realizations according to the SM-STBC
codewords. In (20), y and n represent the 2Nr × 1
equivalent received signal and noise vectors, re-
spectively. No ICI occurs in SM as the columns of
are orthogonal to each other because of Ala-
mouti.
From, x equivalent channel matrices
and for the eth
combination the receiver de-
termines k1 and k2 as follows [18], resulting from
the orthogonality of hl1 and hl2:
(21)
Where and hlb, b= 1,
2, is a 2Nrx1 column vector. The minimum ML
metrics m1l and m2l for k1 and k2 are
(22)
respectively.Summation ,
gives the total ML metric for the eth
combination where and are calculated by
the ML decoder for the lth
combination. The re-
ceiver decide by choosing the minimum antenna
combination metric as for which
. Because of above result
number of ML metric calculations in (18) is re-
duced to 2xM from xM2 yielding a linear decoding
complexity. The last step is the de mapping opera-
tion based on the look-up table used at the trans-
mitter, to recover the input bits
fr
om the determined spatial combination and the
information symbols and . The block diagram
of the ML decoder described above is given in
Fig. 3.
Figure3: Block Diagram of SM-STBC ML receiver
V. SIMULATION RESULT
In figure 4&5, the BER performance of STBC-SM
scheme is compared with Golden code and DSTTD
scheme which are rate-2 (4 symbols being trans-
mitted in 2 time intervals) STBs for 2 and 4 anten-
nas for transmission respectively, at rates of 4 & 6
bits/s/Hz. Though both these systems show ML de-
coding complexity which shows proportionality to
the 4th
power of constellation size. A DSTTD
scheme broadly employs MMSE decoding. Never-
theless, an ML decoder is used for comparing the
performance of the scheme under consideration.
International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013
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Figure 4: BER performance of STBC-SM, Golden code and
DSSTD code for 4 bits/s/Hz spectral efficiency
Figure 5: BER performance of STBC-SM, Golden code and
DSSTD code for 6 bits/s/Hz spectral efficiency
Figure 4 shows that SNR gains of 0.75 db and 1.6
db are offered by the STBC-SM over the DSTTD
scheme and Golden code, respectively at rate of 4
bits/s/Hz, and at the same time it has the same
complexity as that of ML decoding which is equal
to 128. Figure 5, on the other hand, shows SNR
gains of 0.4 dB and 1.5 dB, offered by the STBC-
SM over the DSTTD scheme and Golden codes,
respectively, at a rate of 6 bits/s/Hz, this time
showing a 50% lower decoding complexity equal
to 512.
VI. CONCLUSION
A novel STBC-SM MIMO transmission scheme
with improved spectral efficiency has been pro-
posed in this paper. This scheme enables different
symbols to be transmitted at the same time slot by
selecting a certain no. of transmit antennas from a
set of large no. of antennas. For the current pro-
posed scheme the Rayleigh fading channel was as-
sessed with BER performance as a parameter and
was compared to DSTTD scheme and Golden code
using the MATLAB simulation.
It is seen in the simulation results that the SNR
shows an improvement with the increase in spectral
efficiency. Therefore, a significantly improved
BER performance is offered by STBC-SM when
compared with DSTTD and Golden code scheme.
It can thus be concluded as the improvement in
performance of the new scheme increases with the
increasing transmission rate, so it is a very useful
scheme to be employed for systems with high rate
of data transmission.
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AUTHOR’SBIOGRAPHIES
R.P. Yadav was born in Kanwarpura,Jaipur, in
1962. He received M.Tech. degree from I.I.T, Delhi in in
1987 and Ph.D. degree from university of Rajasthan, Jai-
pur in 2000. He joined as a lecturer in 1987 at Dept. Of
Electrical Engg. MREC, Jaipur. He became Reader in
1993 & professor in 2006. Currently he is working as
vice-chancellor of RTU, Kota, Rajasthan. Dr. Yadav is a
member of IEEE, ISTE, BES (India) & IETE (Jai-
pur).Apart from these he served number of universities
as member of „Board of studies‟, member of „National
Board of Accreditation‟. He has investigated two MHRD
sponsored research projects and has conducted
/organized number of Faculty Development programs,
National & International conferences. He has been ac-
tively involved in research over the last two decades. He
has published more than 55 research papers in reputed
national and international journals and have present his
work at number of national and international confe-
rences. He delivered number of invited talks and key
note addresses at reputed institutes including IITs and
conferences. His research interests are in area of Net-
working, Microwave Communication, MIMO-OFDM
and Error control coding.
Garima Mathur received M.Tech degree in
Electronics and Communication Engineering from the
MNIT Jaipur in 2003, where she is currently working
towards the Ph.D. degree. Her research interests are In-
formation Theory, MIMO-OFDM and Error control cod-
ing.
Sarita Boolchandani received B.E. in Electronics
and communication engineering from the University of
Rajasthan in 2005. She has completed M.Tech. in Digital
Communication from Rajasthan Technical University in
2013. Her research interests are Wireless Communica-
tion, Information theory & Coding.
Shruti Sikarwar received B.Tech. in Electron-
ics and communication engineering from the Rajasthan
Technical University in 2010. She is pursuing M.Tech.
in Communication & Signal Processing from Jaipur Na-
tional University. Her research interests are Wireless &
Digital Communication, Optical Communication.