High-rate STBC-MTCM schemes for quasistatic and block

High-rate STBC-MTCM Schemes for Quasi-static and Block-fading Channels

Rahul Vaze

ECE Dept.,Indian Institute of Science Bangalore-560012INDIA email:in

B.Sundar Rajan

ECE Dept.,Indian Institute of Science Bangalore-560012INDIA

email:bsrajan@in

Abstract—For the case of quasi-static fading channel,high rate Space-Time Trellis Codes have already been constructed by concatenating Multiple Trellis Coded Modulation(MTCM) and Space-Time Block Codes(STBC)called the STBC-MTCM scheme.The focus in all these constructions,was to increase the rate of transmission by using more than one orthogonal design,while retaining the diversity advantage and little attention was paid to increase the coding gain advantage.In this paper, we present a systematic approach by which STTCs can be constructed by STBC-MTCM scheme,which achieve high rate, full diversity and increased coding gain advantage over the existing

codes under certain conditions.

Also we a present a systematic approach,to construct STTCs by STBC-MTCM codes which can achieve any given diversity for the case of block-fading channel.The codes constructed for block-fading channels trade-off the rate of transmission and the number of states of the trellis.

I.I NTRODUCTION

Space-Time Trellis Codes(STTC)have been introduced in[1] to provide improved error performance for wireless systems using multiple transmit antennas.In[2],Alamouti introduced a simple code to provide full diversity for two transmit antennas. In[3],the scheme is generalized to an arbitrary number of antennas and is named space-time block coding.Although a Space-Time Block Code(STBC)provides full diversity and a simple decoding scheme,it does not provide good coding gain,whereas STTC provide full diversity as well as coding gain but at the cost of higher decoding complexity. To achieve additional coding gain,one should concatenate an outer code such as Multiple-Trellis Coded Modulation (MTCM)as deﬁned in[4]with an inner STBC called the STBC-MTCM scheme.

In[5],Alamouti matrix is combined with MTCM to provide more coding gain along with full diversity.Th

e limitation of STBC-MTCM scheme is that the rate of transmission (bits/sec/Hz)gets reduced because the inner block code is at best,a rate-one code and the outer MTCM encoder must have redundancy.In order to enable high data rate via a concatenated STBC-MTCM scheme,the inner block code must be expanded before being concatenated with an outer MTCM encoder.

To increase the transmission rate,the technique adopted in [6]–[9]is to apply some unitary transformations to the original This work was partly supported by the IISc-DRDO program on Advanced Research in Mathematical Engineering through a grant to B.S.Rajan.STBC matrices,so that more number of code matrices are available for transmission,but by using this technique,the difference matrix over all possible pairs of matrices is not full rank.

It is well known that the performance of the STBC-MTCM codes is not directly given by the minimum distance between any two codewords but governed by the distance path weights of the error events or the multiplicities of error events).In this paper,we provide an alternative systematic construction for high rate,full diversity achieving STBC-MTCM code,in which the multiplicities of the error events has been reduced leading to better performance under certain conditions,for the case of quasi-static fading channel.We also provide simulation results to show that under certain conditions our codes outperform the best known codes in literature in terms of coding gain.

In a block-fading channel model the codeword is composed of multiple blocks,the fading coefﬁcients are constant over one fading block,but are independent over block to block.It has been shown in[10]that for block-fading channel,if we code across L quasi-static fading intervals(quasi-static fading interval is the time for which the fading coefﬁcients remain constant),the maximum diversity which we can achieve is L times the diversity which we can achieve in one quasi-static fading interval.

For STBC-MTCM codes,if we have to exploit the block-fading channel to get an arbitrary diversity gain,the necessary condition is that,we have to expand the inner block code in such a way,that the the difference matrix of any two distinct STBC matrices is a full rank matrix.Therefore the technique given in[6],cannot be used to,simultaneously increase the rate of transmission as well achieve any given diversity for STBC-MTCM codes in block-fading channel.In this paper we provide a construction of STBC-MTCM codes which can achieve high rate and any given diversity in the case of block-fading channel.Simulation results are also provided.

The remaining content of the paper is organized as follows: We provide a systematic construction for high-rate,full-diversity STBC-MTCM codes for quasi-static fading channel in Section II.In Section III,a novel approach for construction of high-rate and any speciﬁed diversity achieving STBC-MTCM code for the block-fading channel is given.

01001

101STBC陈迪和

decoder

x k

~x 2

~bits

b bits

b Encoder

MIMO

Channel Rayleigh 0

101x

x k

STBC

encoder

p n t 00001111000011

11000011110011001

100110011001

10011n 1

n t

n n 1

S/P

kb bits

Decoder

P/S

rate k/p

design

MTCM

Fig.1.System Block Diagram

II.Q UASI -STATIC FADING CHANNEL

Space-Time Code Design Criteria:The diversity gain of a

space-time code is deﬁned by the minimum rank of the matrix B (s 1,s 2)=(s 1−s 2)(s 1−s 2)H over all possible distinct codewords s 1and s 2and the coding gain is deﬁned as the minimum of the product of the eigen values of B (s 1,s 2)over all possible pairs of distinct codewords s 1and s 2[1].As in [6],we deﬁne the coding gain distance (CGD)between codewords s 1and s 2as d 2(s 1,s 2)=det (B (s 1,s 2)),where det (B )is the determinant of the matrix B.

Deﬁnition 1:A rate-k/p ,n t ×p design is a n t ×p matrix with entries as linear combination of k complex variables and their conjugates.Restricting the k variables to take values from a ﬁnite subset of C ,we get a Space-Time Block Code (STBC).System Model:The system model we consider is a space time wireless communication system with n t transmit antennas and n r receive antennas.The channel between a transmit and receive antenna is modeled as a frequency non-selective quasi-static Rayleigh fading process,such that channel coefﬁcients remain same in one frame but are independent from frame to frame and from antenna to antenna.

Let the inner STBC be obtained from a rate-k/p ,n t ×p design X .If the required rate of transmission is b bits/sec/Hz,then the transmitter takes kb bits as input and each of the k streams of b bits undergo Trellis Coded Modulation as shown in Fig.1.The output of the MTCM scheme is fed to the STBC encoder.STBC encoder which contains the design matrix X ,takes these k symbols and forms

a n t ×p codeword matrix.Each branch of the trellis represents one STBC matrix.This matrix represents the low pass representations of the signal to be transmitted by n t antennas for the next p symbol durations.The system block diagram is shown in Fig.1.At the receiver end,the Space Time block decoder followed by a Viterbi decoder can be used to decode the received signals as given in [5].

Code Design Requirements:For designing a rate b bits/sec/Hz STBC-MTCM code,2pb transitions should come out of each branch of the trellis,but to avoid a catastrophic code,the number of different STBC matrices required for mapping the transitions in the trellis are at least 2pb +1(The catastrophy which we are concerned here,is the catastrophy which can be generated by faulty mapping of the trellis branches and not the catastrophy of the MTCM encoder,as

given in [6]).

For simplicity,we take n t =2,n r =1and our STBC

to be the Alamouti matrix x 0x 1

−x ∗1x ∗0

(k =2,p =2).

The number of possible Alamouti matrices over a signal constellation of size 2b is 22b ,therefore 22b more Alamouti matrices are required to design a rate b bits/sec/Hz STBC-MCTM non catastrophic code.If we double the size of signal constellation,the number of all possible Alamouti matrices is 22b +2,out of which only half are required to construct a rate b bits/sec/Hz non catastrophic STBC-MTCM code.

Code Construction:In our construction,we will double the signal constellation size to get the required number of Alamouti matrices.To maximize the coding gain,we do set partitioning in terms of CGD,to choose the required number of matrices from the expanded set of STBC matrices,which are optimal in the sense of CGD.

S 0

S S 00 01 S S 10 S 11

S 000S 001 S 011 S 101 S 010 S

100

S 110 111minimum CGD

内蒙古民族高等专科学校4

1600,22

02,20

11,33

13,31

03,21

10,32

12,30

01,23

Fig.2.Set partitioning for QPSK;the numbers at the leaves represent the indexes of the symbols in the space-time block code.

S 0

S 1

S 00

S 01

S 10

S 11

S 001S 010110S S 100S 101

S 111S 011S 000

0000

S 0001S 0010S 0011S 1111S 1110S 1101S 1100S 0100S 0101S 0111S 0110S 1000S 1010S 1011S 100100441440

22662662

24602064

11551551

33773773

13571753

31753571

2367276303470743

21652561

10541450

32763672

30743470

CGD minimun 02460642

01450541

1656121616

1.3740.34Fig.3.Set partitioning for 8-PSK;the numbers at the leaves represent the indexes of the symbols in the space-time block code.

In general,this technique of signal set expansion,set parti-tioning over a M -PSK (M =2b )constellation and choosing the restricted set of STBC matrices required for transmission,can be explained as follows:

First Step:Take all possible Alamouti matrices over M -PSK constellation and call this set of matrices A 1and then do the set partitioning of the matrices in the set A 1,in terms of CGD to form sets A 11,A 12,...,A 1r ,with each A 1i having 22b /r number of elements ∀i,i =1,2,...,r ,where r is chosen according to the design requirement of the code.

Second Step:Rotate the M -PSK constellation by angle π/M and then take all possible Alamouti matrices over this rotated constellation and call this set of matrices A 2.Similarly do the set partitioning of the matrices of the set A 2in terms of CGD,to form sets A 21,A 22,...,A 2r as above.

By using both the sets A 1and A 2,we choose 22b +1Alamouti matrices which are optimal in the sense of CGD,required for the construction of rate b bits/sec/Hz STBC-MTCM code,which are full rank.

Code Design Rules:In our scheme,we assign a constituent space-time block code to all transitions from a state.The adjacent states are typically assigned to one of the other constituent space-time block codes from the set A 1or A 2.The parallel transition branches are assigned STBC matrix from one of the A ij where i =1,2and j =1,2,...,r .Similarly,we can assign the same space-time block code to branches that are merging into a state from either A 1or A 2.It is thus assured that any path that diver

ges from (or merges to)the correct path differs by rank 2.In other words,every pair of codewords diverging from (or merging to)a state achieves full diversity because the pair is from the same orthogonal code.Design Examples:Now we present several design exam-ples of our new proposed code with rate 1bit/s/Hz and 2bits/sec/Hz using QPSK and 8-PSK constellations.The proposed simple design rule is used to construct the STBC-MTCM codes that achieve full diversity.MTCM with multi-plicity of 2is used as an outer encoder,and thus 4and 16outgoing transitions are needed to achieve the desired code rate of 1bit/sec/Hz and 2bits/sec/Hz respectively.

Figs.4and 5,show the new 2-state and 4-state 1bit/sec/Hz and 2bits/sec/Hz space-time codes respectively.For the rate 1bit/sec/Hz and 2bits/sec/Hz codes we use Alamouti matrices over QPSK and 8-PSK constellation respectively for mapping the branches of the trellis from Figs.2and 3.

010 S S 011

001S 000Fig.4.A two state code;rate 1bit/sec/Hz using QPSK or 2bits/sec/Hz using 8-PSK.

001

011S 010

000S 001S 000S 010S

011Fig.5.A four state code;rate 1bit/sec/Hz using QPSK or 2bits/sec/Hz using 8-PSK.

Coding Gain Analysis:We calculate distance spectrum for all our codes and compare that with the best known codes in Super Orthogonal Space-Time Trellis Codes (SOSTTC)[6].We tabulate,the CGD of all our codes in Table I to Table VI and compare them with the appropriate codes as in [6].

Deﬁnition 2:In a trellis,two code sequences constitute an

error event of length l ,if they start from the same state and rejoin at some other state for the ﬁrst time after l intervals.If the number of input bits is equal to b and the memory of the MTCM encoder is m (number of states of trellis equal to 2m ),then the minimum value of l is m/b .We denote this smallest value of l by P .Effective length of a MTCM code,ν,is the minimum number of distinct symbols between any two codewords and the maximum achievable effective length is given by ν= m/b +1as in [11].

In Table I,we tabulate the CGD on parallel paths and minimum possible CGD on any other path.For parallel transitions (P =1),one can get the CGD from Figs.2or 3.For paths other than the parallel transitions,we consider two codewords diverging from state zero and re-merging after P transitions to state zero.For these paths,the CGD is calculated as the determinant of the difference matrix,as already deﬁned.For 2-state codes,P is equal to 2and the number of paths coming back after P =2,for rate 1bit/sec/Hz and 2bits/sec/Hz codes are equal to 4and 64respectively.In Table III and IV,we tabulate the CGD of these paths and compare them with the appropriate code of [6],for 2-state rate 1bit/sec/Hz and 2bits/sec/Hz code respectively.For 4-state codes,P =3and the number of paths coming back after P =3,for rate 1bit/sec/Hz and 2bits/sec/Hz codes are 8and 512respectively.In Tables V and VI,we tabulate the CGD of these paths and compare them with the equivalent code of [6],for 4-state,rate 1bit/sec/Hz and 2bits/sec/Hz code respectively.

Table III to Table VI,give the distance spectrum,of all our codes.It is enough to calculate the CGD’s upto minimum possible value of l i.e P ,as our codes are non-catastrophic.From Table I,except for our 2-state rate 2bits/sec/Hz code,the minimum CGD of our codes is greater than or equal to the minimum CGD of the comparable SOSTTC.Clearly from Tables III-VI,the distance spectrum our codes is better than the distance spectrum for the SOSTTC.The resulting performance improvement is discussed in the following subsection.

TABLE I

CGD VALUES FOR DIFFERENT CODES

Rate in bits/sec/Hz

min det(A)

parallel CGD

1(Fig.5)64641(Fig.6)144642(Fig.5)10.05162(Fig.6)

27.04

TABLE II

C OMPARISON OF CG

D VALUES

Figure No.of Rate minimum minimum CGD States (bits/sec/Hz)

CGD in SOSTTC

521644852210.0516********

TABLE III

CGD VALUES FOR ALL PATHS WITH P=2FOR RATE 1BIT /SEC /H Z ,2STATE

CODE

Coding Gain No.of paths No.of paths Distance in our code

in SOSTTC

644-48公安海警

-4

Simulation Results:In this subsection,we provide simula-tion results for our new code design using two transmit and one receive antenna.We compare our results with the SOSTTC for

TABLE IV

CGD VALUES FOR ALL PATHS WITH P=2FOR RATE 2BITS /SEC /H Z ,2

STATE CODE

Coding Gain No.of paths No.of paths Distance in our code

in SOSTTC

10.0516-24-643632-78

TABLE V

CGD VALUES FOR ALL PATHS WITH P=3FOR RATE 1BIT /SEC /H Z ,4STATE

CODE

Coding Gain No.of paths No.of paths Distance in our code

in SOSTTC

1448-128

same number of transmit and receive antennas and for same rate of transmission.In all simulations,a frame consists of 130transmissions out of each transmit antenna.Figs.6and 7,shows the frame error probability results versus signal-to-noise ratio (SNR)for the code given in Figs.4and 5respectively.Our proposed code for rate 1bit/sec/Hz for 2or 4-states,outperforms similar SOSTTC by nearly 0.5dB,also our proposed 2bit/sec/Hz,4-state code,gives a better performance of nearly 0.25dB than the similar code in [6],but rate 2bit/sec/Hz,2-state code in [6]performs better than our code.If we see the simulation results for the 2bits/sec/Hz,2state code,we ﬁnd out that the performance is not as degraded as indicated by the decrease in minimum CGD,because our

fading

Fig.fading III.B LOCK -FADING CHANNEL

System Model:The system model we consider remains the

TABLE VI

CGD VALUES FOR ALL PATHS WITH P=3FOR RATE 2BITS /SEC /H Z ,4

STATE CODE

Coding Gain No.of paths No.of paths Distance in our code

in SOSTTC

27.04128-48-38464256128120

128

same as already shown in Fig.1,except that the channel we consider is a block-fading channel.In block-fading channel the channel coefﬁcients remain same in one block of quasi-static fading interval,but are independent from block to block and from antenna to antenna.

Design Criteria for Block-Fading Channel:Let C be a code for the channel with n t transmit and n r receive antennas.We assume that the code C has codewords spread over K quasi-static fading blocks,the length of each quasi-static block being p .Thus,the codeword c =[c [1]c [2]...c [K ]],where c [i ]is the part of codeword corresponding to i -th fading block,is a n t ×pK matrix.The generalized diversity and product distance criteria for space-time codes over MIMO block-fading channels are as follows.Maximize the transmit diversity advantage

d =

K i =1

d i =

K i =1

rank (c [i ]−e [i ])(1)

(2)

trellis is least νon each in (1),diversity distinct should design a STBC-MTCM code that achieves diversity gain of νD ,the effective length of the trellis should be atleast ν.

Code Design:For simplicity,we take n t =2,n r =1and our STBC to be the Alamouti matrix.To get diversity νD and rate b bits/sec/Hz,for STBC-MTCM scheme in block-fading channel,we will use a MTCM with effective length νand the same construction for increasing the rate of transmission for the STBC-MTCM code for quasi-static fading channel,of doubling the signal set and set partitioning the set of all possible Alamouti matrices in terms of the CGD.This guarantees that the difference matrix of any two such Alamouti matrices is a full-rank matrix.

Code Design Rules:In our scheme,we assign a constituent STBC to all transitions from a state.The adjacent states are typically assigned to one of the other constituent STBC from the set A 1or A 2.We avoid parallel transitions in the trellis to

get extra diversity gain.By our construction,the difference of any two distinct Alamouti matrices is a full rank matrix,then from the sum of rank criteria,we are guaranteed of at least,a diversity gain equal to 2ν,if the effective length of the trellis is νas diversity gain given by Alamouti matrix is 2.

Design Examples:We present design examples of our proposed code.Figs.8and 9,give the design of diversity 4STBC-MTCM codes,for rate 1bit/sec/Hz and 2bits/sec/Hz,respectively.In both the examples,parallel transitions have been avoided to get extra diversity.

811事件

For designing rate 1bit/s/Hz and 2bits/sec/Hz diversity 4STBC-MTCM codes,we use Alamouti matrices over QPSK and 8-PSK constellations respectively for mappings branches of the trellis from Figs.2and 3.For the design examples,MTCM with multiplicity of 2is used as an outer encoder,and thus 4and 16outgoing transitions are needed to achieve the desired code rate of 1bit/sec/Hz and 2bits/sec/Hz respectively.

20 02 22 0031 13 33 11

11 33 13 31

00 22 02 20Fig.8.A four state code;rate 1bit/sec/Hz using QPSK set partitioning as shown in Fig.2

26 6202 4624 6020 6400 4400 4404 4022 6604 4022 6600 4400 4424 6020 6402 4606 4224 6020 6402 4606 4204 4022 6600 4400 4400 4400 4422 6604 4006 4226 6220 6424 6013 57 17 5331 75我型我秀2007

35 7111 5515 5133 7737 73

31 7533 7737 7311 55

15 5135 7113 57 17 53

15 5133 7737 7311 5531 7535 7113 57 17 5315 5111 5537 73

33 7717 5313 57 35 7131 75

17 5313 57 31 7515 5111 5537 7335 7133 7720 6424 6006 42

02 4622 6604 4000 4400 44

33 7737 7335 7131 7517 5311 5515 5113 57 31 7513 57 37 73

33 7715 5135 7117 5311 55

00 44 04 40 22 66 26 62 02 46 06 42 24 60 20 6406 4202 4620 6424 6000 4400 4422 6604 4000 4406 4202 4604 4000 4420 6424 6022 66

11 55 15 51 33 77 37 73 13 57 17 53 31 75 35 71Fig.9.A sixteen state code;rate 2bits/sec/Hz using 8-PSK set partitioning as shown in Fig.3

Note :The codes which we present,trades-off the rate of transmission and the number of states of the trellis (decoding complexity). achieve diversity 4,for rate 2bits/sec/Hz,we have to increase t

he number of states to 16as shown in Fig.9.Therefore,ﬁxing the diversity gain,increase in rate is possible only at the cost of increased decoding complexity.Coding Gain Analysis:For our codes ν=2,therefore the number of quasi-static fading blocks across which we are able to do coding is 2.The diversity gain of codes given in Figs.8and 9is 4.For any state in our trellis,the diverging branches are mapped from set S 00or S 01and converging branches are mapped from set S 0,therefore from (2),the coding gain is

given by the product of the intraset minimum CGD of matrices S 00and S 0or the product of the intraset minimum CGD S 01and S 0,raised to power of 1/4,where S 00,S 01and S 0as given in Figs.2and 3for QPSK and 8-PSK case respectively.The coding gain for rate 1bit/sec/Hz,diversity 4,STBC-MTCM code as given in Fig.8,is (16×16)1/4which is 2and for rate 2bit/sec/Hz,diversity 4STBC-MTCM code as given in Fig.9,the coding gain is (4×1.37)1/4which is 1.53.Fig.10,shows the frame error probability results versus signal-to-noise ratio (SNR)for the codes given in Figs.8and 9.Clearly,it can be seen from the simulation results that,our codes achieve diversity 4.

Fig.1receive antenna,block fading system

R EFERENCES

[1]Vahid Tarokh,N.Seshadri and A.R.Calderbank,“Space-Time block

codes for high data rate wireless communication:Performance criterion and code construction,”IEEE Trans.Inform.Theory,vol.44,pp.744-765,Mar 1998.

[2]S.M.Alamouti,“A simple transmit diversity technique for wireless

communication,” Select.Areas in Commun.,vol.16,no.8,pp.1451-1458,Oct.1998.

[3]Vahid Tarokh,H.Jafarkhani and A.R.Calderbank,“Space-Time block

codes from orthogonal designs,”IEEE Trans.Inform.Theory,vol.45,pp.1456-1467,July 1999.Also “Correction to “Space-time block codes from orthogonal designs”,”IEEE Trans.Inform.Theory ,vol.46,no.1,p.314,Jan.2000.

[4] D.Divsalar and M.K.Simon,“Multiple Trellis Coded Modula-tion,” Comm.,vol.36,no.4,pp 410-419,April 1988.[5]S.Alamouti,V .Tarokh,and P.Poon,“Trellis-coded modulation and

transmit diversity:Design criteria and performance evaluation,”in Proc.IEEE Int.Conf.Universal Personal Communications (ICUPC-98),vol.2,1998,pp.917-920.

[6]Hamid Jafarkhani,and Nambi Seshadri,“Super-Orthogonal Space Time

Trellis Codes,”in IEEE Trans.Inform.Theory ,vol.49,no.4,APRIL 2003.

[7]S.Siwamogsatham and M.P.Fitz,”Improved high rate space-time codes

via orthogonality and set partitioning,”in Proc.IEEE Wireless Commu-nications and Networking Conf.(WCNC),Mar.2002.

[8]—”Improved high rate space-time codes via concatenation of expanded

orthogonal block code and M-TCM,”in IEEE International Conference on Communications (ICC),vol.1,Apr.2002,pp.636-640.

苗颖

[9]—”Improved high-rate space-time codes via expanded STBC-MTCM

constructions,”in Proc.IEEE Int.Symp.Information Theory (ISIT),Lausanne,Switzerland,June/July 2002,p.106.

[10]Hesham El Gamal and A.Roger.Hammons,“On the Design of Algebraic

Space-Time codes for MIMO Block Fading Channels.”in IEEE Trans.Inform.Theory ,vol.49,no.1,p.314,Jan.2003.

[11]S.H.Jamali and Tho Le-Ngoc,“Coded Modulation Techniques For Fad-ing Channels,”Kluwer Academic Publishers 1994

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