STBC-SM BASED RELATIVE ESTIMATION OF GOLDEN AND DSTTD CODES

International Journal of Modern Electronics and Communication Engineering (IJMECE) ISSN: Awaited Volume No.-1, Issue No.-1, March, 2013

RES Publication © 2012 Page | 12 http://www.resindia.org

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,

[email protected]

Sarita Boolchandani ShrutiSikarwar

M.Tech JEC Kukas, RTU, Kota, M.Tech student, JNU, Jaipur

[email protected] [email protected]

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



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-



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



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:



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)



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-



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.



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.

STBC-SM BASED RELATIVE ESTIMATION OF GOLDEN AND DSTTD CODES

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Transcript of STBC-SM BASED RELATIVE ESTIMATION OF GOLDEN AND DSTTD CODES