INTRODUCTION
In order to achieve reliable digital communications over fading channels, high
signal energies and large bandwidth expansion factors should be required for
time diversity signaling. Schlegel and Costello (1989) was used the Chernoff
bounding technique to obtain performance bounds for bandwidth efficient trellis
codes on fading channels with various degrees of side information and used the
effective length and the minimum squared product distance to replace the minimum
free squared Euclidean distance as code design parameters for fading channels
with a substantial multipath component.
TCM (Ungerboeck, 1982) is one of the coded modulation techniques used in digital
communications. It combines the choice of a modulation scheme with that of a
convolutional code together for the purpose of gaining noise immunity over encoded
transmission without expanding the signal bandwidth or increasing the transmitted
power (Garello et al., 2002). The main advantage
of TCM technique, which makes it suitable scheme for transmitting digital sequences
over bandlimited channels, is providing good coding gain without sacrificing
any bandwidth. TCM can be viewed as a combined coding and modulation technique
wherein modulation is embedded into the encoding process and is designed in
conjunction with a rate n/(n+1) convolutional code (Garello
et al., 2002). Trellis codes specifically constructed for fading
channels was presented by Schlegel and Costello (1989)
and an efficient method to evaluate the Chernoff bound on the event error probability
of these codes, their performance is analyzed for fading channels with different
degrees of side information. numerical results on event error probability demonstrate
the importance of the effective length as a code design parameter for fading
channels. A 4state 8PSK TCM code schemes for both speech and data transmission
over mobile fading channels is addressed by Jamali and LeNgoc
(1991), the design rules is based on that the minimum product of the squared
branch distances along the shortest error event paths is maximized. Simulation
results for light shadowed Rician fading channel showed that this code has better
performance than the other 4state 8PSK TCM schemes for both data and speech
bit error rates. A coordinate interleaving (Boulle and Belfiore,
1992) based TCM code construction scheme designed for flat fading channels
was introduced by Jelicic and Roy (1994). Due to finite
interleaving and imperfect CSI effects, a significant portion of their coding
gain is preserved, It was demonstrated that although coordinate interleaving
schemes suffer relatively more impairments than corresponding symbol interleaving
schemes. Code performance including the effects of finite interleaving and imperfect
Channel State Information (CSI) was studied. the inherent coding gain of proposed
coordinate interleaving based schemes are substantiated by cutoff rate computations.
A 4PSK 16state spacetime coded OFDM scheme was proposed by Agrawal
et al. (1998) for providing high datarate delaysensitive wireless
communication over wideband channels and its performance was compared to the
RS coded OFDM scheme. Simulation results showed that the proposed scheme is
capable of robust reliable transmission at relatively lower SNRs in a variety
of delay profiles. A simple spacetime coded OFDM transmitter diversity technique
for frequency selective fading channels was presented by Lee
and Williams (2000). The proposed technique utilizes OFDM to transform frequency
selective fading channels into multiple flat fading subchannels on which spacetime
coding is applied. In slow fading channels, the proposed transmitter diversity
system achieves diversity gain equivalent to that of the optimal Maximal Ratio
Combining (MRC) receiver diversity system. A twobranch transmitter diversity
system is implemented without bandwidth expansion and with a small increase
in complexity beyond that of a conventional OFDM system. Simulations verify
that the twobranch transmitter diversity system achieves a diversity gain equivalent
to that of the optimal MRC receiver diversity system. A trellisstructured STC
designing scheme in OFDM systems for frequencyselective fading channels was
proposed by Lu and Wang (2000), the Pairwise Error Probability
(PEP) expression show that STCOFDM systems can potentially provide a diversity
order as the product of the number of transmitter antennas, the number of receiver
antennas and the frequency selectivity order and also proposed that the large
effective length and the ideal builtin interleaving are two most important
elements in designing STCOFDM system. Due to efficiently exploit both the spatial
diversity and the frequencyselectivefading diversity, the simulation results
demonstrated the significant performance improvement of the proposed STC’s
over the conventional spacetime trellis codes.
The MIMOOFDM systems can achieve significant error rate performance improvement
by exploiting spatial and multipath diversities. The spatial diversity can be
achieved with spacetime coding at the transmitter and signal combining at the
receiver, the multipath diversity can be achieved with channel coding through
subcarriers in case of OFDM modulation. OFDM converts a frequencyselective
fading channel into parallel flatfading subchannels, thereby simplifying channel
equalization and symbol decoding. However, OFDM’s performance suffers from
the loss of multipath diversity and the inability to guarantee symbol detectability
when channel nulls occur. To achieve reliable wireless transmission over a slowly
varying MIMO channel, Alamoutibased spacetime BitInterleaved Coded Modulation
(BICM) with Hybrid Automatic Repeat reQuest (HARQ) transmission scheme is proposed
by Wang et al. (2009a) to increases the efficiency
of HARQ packet transmission by exploiting both the spatial and time diversity
of the MIMO channel, which uses the full diversity of Alamouti STC and the added
gain of the packet combining. An OFDM transmissions scheme using Linear Constellation
Precoded (LCP) to achieves multipath diversity was introduced (Liu
et al., 2003) for frequencyselective fading channels. Exploiting
the correlation structure of subchannels and choosing properly system parameters,
an optimal subcarrier grouping to partition the set of subchannels into subsets
was performed. Relying on subcarrier grouping, the LCPOFDM system was converted
into a set of GLCPOFDM subsystems and then designed LCPs for each subsystem.
Subcarrier grouping enables the maximum possible diversity and coding gains
while reducing the decoding complexity greatly and simplifying the precoder
design. For multicell multiantennas networks with uplink training, in order
to mitigate corrupted uplink training and thus increase achievable rates of
downlink transmission, a precoding techniques was proposed by Wang
et al. (2009b), this precoding is the optimal solution of an optimization
problem of the meansquare error of signals and the meansquare interference.
A SpaceTimeFrequency (STF) coding scheme achieving spatial and multipath diversities
for MIMOOFDM transmissions over frequencyselective Rayleigh fading channels
was proposed by Liu et al. (2002). In order to
simplify the design, convert the complex STF codes design into simpler GSTF
designs per group by incorporating subchannel grouping and choosing appropriate
system parameters. both GSTF block codes and GSTF trellis codes was constructed
based on the derived design criteria, which achieves the full diversity gain
with low decoding complexity, this equals the product of the number of transmit
and receive antennas times the channel length. A joint ErrorControl Coding
(ECC) and ComplexField Coding (CFC) scheme to further increase the multipath
diversity for MIMO spacetime transmission over flat or frequencyselective
fading channels was proposed by Wang et al. (2003).
In this scheme, trellis encoding through OFDM subcarriers was used with serially
concatenated outer convolutional code and inner LCPSTBC (SCLCPSTBC). Relying
on OFDM transmitters to convert an FIR MIMO channel to correlated parallel flat
fading MIMO channels, the CFC scheme for SISO flat fading channels was adapted
to spacetime flat fading MIMO channels. The diversity achieved by the joint
system with transmit antennaswitching is the product of the free distance of
the ECC, the complexfield encoder size and the number of receive antennas.
Exploring the lattice sphere packing representation theory (Oggier
and Viterbo, 2004), Sphere Decoding (SD) algorithm (Damen
et al., 2000) was proposed for the VBLAST multiantenna system and
the algebraic SpaceTime (ST) codes over the Rayleigh fading channel. The simulation
results shows that the algorithm obtains the MaximumLikelihood (ML) performance
with a computational complexity, It is does not depend on the constellation
size. Hence, a very high spectral efficiency could be obtained along with ML
performance. The simulation results also shows that the limitations of the VBLAST
detection algorithm over the uncoded system in taking advantage of the receive
diversity. The algebraic ST codes can obtain the full diversity in a multiantenna
system without adding any redundancy and perform the ML decoding with low computational
complexity.
It is known that by employing SpaceTimeFrequency Codes (STFC) to frequency
selective MIMOOFDM systems, all the spatial, temporal and multipath diversity
can be exploited. There exists STFBC designed using orthogonal designs with
constellation precoder to get full diversity (Liu et
al., 2002) for more than 2 transmit antennas, STFBC of rate1 full diversity
cannot be constructed using orthogonal designs due to orthogonal designs of
rate1 exists only for 2 transmit antennas. A rate1 (complex symbols per channel
use) STFBC scheme of rate1 for 4 transmit antennas MIMOOFDM systems with double
symbol decoding designed using quasiorthogonal designs combined with CIOD was
presented by Gowrisankar and Rajan (2005), whereas all
known schemes provide only rate 3/4 complex symbols per channel use. This is
the first scheme using QuasiOrthogonal Design to construct STFBC for MIMOOFDM.
the conditions on the signal sets used for the variables of the QuasiOrthogonal
design for fulldiversity are obtained and simulation study shows that the BER
performance is at least as good as that of the existing comparable scheme.
The STBC from CIOD offers advantages of fulldiversity and singlesymbol decidability
(Khan and Rajan, 2006), symmetric structured CIOD are
constructed by incorporating two same codewordstructured generalized linear
complex orthogonal designs (GLCOD). STBC scheme from Symmetric structured CIOD
for a quasistatic frequencynonselective i.i.d. Nakagamim fading channel was
presented by Lee et al. (2009) and the performance
analysis of average error rate (e.g., average Symbol Pairwise Error Rate (SPER),
SymbolError Rate (SER), etc.), outage capacity, Information Outage Probability
(IOP) and both average SERbased and IOPbased asymptotic diversity orders was
presented. Tight union upper and lower bounds on the SER was derived from an
accurate closedform formula for the SPER. Closed form expressions for the outage
capacity was provided using Gaussian and Gamma approximations. Accurate closedform
approximations for the IOP was derived using high SignaltoNoise Ratio (SNR)
and momentmatching approximation techniques. The proposed STBC offering fulldiversity
was proved by SER and IOPbased asymptotic and instantaneous diversity orders.
The STBC using CIOD allow singlecomplex symbol decoding and offer higher data
rates than traditional orthogonal STBC. An equivalent channel representation
for CIOD codes enabling their decoding readily over MIMO channels was presented
(Dao and Tellambura, 2008), A general ML metric is derived,
enabling the computation of Symbol PairWise Error probability (SPEP) and Union
Bound (UB) on SER. The UB thus can be used to accurately predict and optimize
the performance of CIOD codes. a signal design combining signal rotation and
power allocation is presented for constellations with uneven powers of real
and imaginary parts. the union bound can be used to accurately analyze the performance
of CIOD codes and moreover, to optimize the signal rotation for arbitrary constellation.
For traditional SpaceFrequency (SF) coded OFDM systems, in order to exploit
the maximum achievable diversity order, we should be needed to design spacetime
trellis codes with high trellis complexity. A concatenated Trellis Coded Modulation
(TCM) codes with STBC as an efficient SF coding scheme in coded OFDM system
was proposed for highspeed transmission over wireless links (Gong
and Letaief, 2003). The analytical expression for the pairwise probability
of the proposed system is derived in slow, space and frequencyselective fading
channels. The design criteria of TCM codes used in such SF OFDM systems are
derived. Simulation results shows excellent performance results in the space
and frequencyselective fading channels, It is also shown that the proposed
SF codes with low trellis complexity can easily exploit the full diversity order
provided by the fading channel, as well as the excellent outage capacity properties
of SF coded OFDM over multipath fading channels. An TCM concatenated with Differential
SpaceTime Block Codes (DSTBC) designing scheme was proposed in frequencyflat
Rayleighfading channels with and without perfect interleaver (Tarasak
and Bhargava, 2004). The design criteria of the proposed scheme are effective
code length over span two symbol intervals and minimum productsum distance
over span two symbol intervals, which are exactly the same as that of TCMSTBC
in perfectly known channels. Based on the design criteria, several rate2/3
8PSK systematic Ungerboeck’s TCM schemes (Ungerboeck, 1982) are found
by computer search and shown to outperform the optimal codes design for AWGN
channel and Rayleighfading channel without transmit diversity. The presented
codes can be used with STBC in perfectly known fading channels. However, the
computational complexities of a ML sphere decoder (Damen
et al., 2000) used with LCPSTBC is very high and a turbo decoder
used with SCLCPSTBC leads to a search for coded modulation techniques with
lower decoding complexities. Such efficient technique with simple Viterbi decoding
is a concatenated outer trellis coded and inner SpaceTime Block Coded (TCSTBC)
OFDM (Gong and Letaief, 2003). It is shown by Gong
and Letaief, 2003 that TCSTBC has a superior performance compared to other
trellis coded MIMOOFDM systems (Agrawal et al., 1998;
Lu and Wang, 2000).
On the other hand, coordinate interleaving (Jelicic and
Roy, 1994; Oggier and Viterbo, 2004) is a powerful
technique providing diversity by extending the considered signal constellation.
A spacetimefrequency block codes (STFBC) for MIMOOFDM transmissions over
frequency selective Rayleigh fading channels was presented (Jagannadha
Rao et al., 2004). When the number of nonzero taps of the Channel
Impulse Response (CIR) is equal to two, symbolbysymbol decoding can be performed
on these codes and these codes have reduced complexity for more than two channel
taps. The STBC from CIOD and Orthogonal Designs (ODs) have been attracting wider
attention due to their amenability for fast (singlesymbol) ML decoding and
fullrate with fullrank over quasistatic fading channels. these STBC codes
are instances of singlesymbol decodable codes. Linear STBC (Khan
and Rajan, 2006) allow singlesymbol ML decoding (not necessarily fulldiversity)
over quasistatic fading channelscalling them singlesymbol decodable designs
(SDD). The class SDD includes ODs and CIODs as proper subclasses. a class of
those SDD that Fullrank SDD (FSDD) offering fulldiversity are characterized
and classified. For square designs, the maximal rate for square FSDDs was deriveed.
For nonsquare designs, generalized CIOD (a superset of CIODs) are presented
and analyzed. For rapidfading channels, an equivalent matrix channel representation
allows the results of quasistatic fading channels to be applied to rapidfading
channels, the rate of singlesymbol decodable STBC are independent of the number
of transmit antennas and inversely proportional to the blocklength of the code.
The CIOD for two transmit antennas is the only STBC that is singlesymbol decodable
over both quasistatic and rapidfading channels. Importantly, the CIOD has
a single symbol decodability feature (Khan and Rajan, 2006)
ensuring low decoding complexities.
In this study, an outer trellis code concatenated with an inner CIOD OFDM system
with low decoder complexity is proposed. This trellis code CIOD system combines
a STBC, a coordinate interleaver and a trellis code to enhance the MIMOOFDM
performance. The analysis of trellis code CIOD codewords pairwise error probability
(PEP) shows that trellis code CIOD provides full spatial and high multipath
diversities and then extend the two symbol interleaved (Tarasak
and Bhargava, 2004) case of TCSTBC (Gong and Letaief,
2003) to a symbol interleaved case of TCSTBC. Compared to the two symbol
interleaved case (Gong and Letaief, 2003), the results
of TCSTBC analysis show that the symbol interleaving doubles the achieved diversity
of TCSTBC. Furthermore, the results of trellis code CIOD analysis show that
the trellis code CIOD achieves diversity double that of the symbol interleaved
TCSTBC. Due to perform rotating and coordinate interleaving operations on a
trellis codeword symbols and then perform spacetime block coding operation
in trellis code CIOD, an additional signal space diversity is achieved, which
doubles the multipath diversity achieved by symbol interleaved TCSTBC. Hence,
the trellis code CIOD diversity is the same as that of SCLCPSTBC (Wang
et al., 2003) while preserving the low ML Viterbi decoding complexities,
the diversity is four times higher than the twosymbol interleaved TCSTBC (Gong
and Letaief, 2003). Subsequently, the trellis code CIOD trellis design rule
is derived from analysis of codeword error probability and further applied to
optimize the performance of trellis code CIOD for {4, 8, 16, 32}state rate2/3
8PSK trellis codes. the simulation results show that the codeword decision
error rate (CER) performances of trellis code CIOD is excellent than that of
the benchmark systems.
SYSTEM MODEL
Here, we describe the channel model and the encoding and decoding operations of the proposed trellis code CIOD OFDM system.
Channel model: Assumed that the channel conditions remain constant during two consecutive OFDM frames transmission. the frequency domain fading coefficients H_{1}[α(k)] and H_{2}[α(k)] represent effects from the first and second transmit antennas during the transmission of Y(k), respectively and denote H_{μ}[α(k)] as K_{k,μ} (μ = 1,2). Let, r_{k,i} be the received symbol from α(k)th subcarrier of the ith OFDM codeword symbol (I = 1,2) and the subchannel noise variables n_{k,i} are independent and identically distributed (i.i.d.) zeromean complex Gaussian random variable with variance 1/2N_{0} per dimension, wherein k = 0,1,...,K1 and i = 1,2 the MIMOOFDM codeword transmission can expressed as:
where, R(k) = (r_{k,1} r_{k,2})^{T}, H(k) = (H_{k,1} H_{k,2})^{T} and N(k) = (n_{k,1} n_{k,2})^{T} and (•)^{T} is the transpose operation. Assumed that a trellis code CIOD OFDM codeword Y = {Y(0),Y(1),...,Y(k1)} transmiting over K subcarriers is partitioned into 1/2K CIOD groups. The corresponding reception and the additive Gaussian noise can be expressed as R = {R(0), R(1),...,R(k1)} and N = {N(0), N(1),...,N(k1)}, respectively.
Encoding operation: Figure 1 shows the encoder block diagram of trellis code CIOD OFDM for two transmit antennas, the source bits are trellis encoded at rate2/3 and mapped to trellis codewords of length 2K with symbols x_{k} from 8PSK signal constellation. and then, each 8PSK symbol x_{k} is rotated by θ,
A vector of rotated symbols:
is perfectly coordinate interleaved. A proper coordinate interleaver should
be employed to achieve maximum diversity.
is the kth element of vector ,
the proposed coordinate interleaver performs the following operations:

Fig. 1: 
Transmitter in the trellis code CIOD OFDM system 
for k = 0,...,K1. Re{•} and Im{•} represent the real and imaginary parts of a complex symbol, respectively and (mod 2K) takes modulo 2K operation. Define a CIOD group for two transmit antennas and two OFDM subcarriers as:
After performing spacetime block encoding of coordinate interleaved symbols
,
,
and ,
Eq. 5 consists of two SpaceTime Block (STB) matrices (Alamouti,
1998). From coordinate interleaving Eq. 4, have:
where, k =0,...,1/2K1. Define:
the CIOD group can be expressed as {Y(k), Y(k+1/2K)} from Eq. 5. According to onetoone mapping relation between k and OFDM subcarriers (k = 0,1,...,K1), each STBC matrix Y(k) of the CIOD group modulates the α(k)th OFDM subcarrier, α(k) is the channel interleaved position of k. the first and second columns of Y(k) are transmitted from the first and second antennas, respectively and the first and second rows of Y(k) are transmitted by two consecutive OFDM frames.
Decoding operation: Assumed that the receiver have perfectly Channel State Information (CSI) H = {H(0), H(1),...,H(K1), the decoding metric of the trellis codeword X = (x_{0},x_{1},...,x_{2K1}) can be expressed as:
where, the CIOD group decoding metric:
According to Eq. 2 and Eq. 4, the ML decoding rule can be expressed as X minimization of Eq. 8. Due to the elements of N are i.i.d., complex Gaussian random variables, each term in the righthand side of Eq. 8 can be expressed as:
and have the metrics related to the symbols ,
,
and
:
where, i = 1,2 and ξ = k,k+1/2K. In Eq. 11, the estimation expressions of rotated and coordinate interleaved trellis codeword symbols:
and the equivalent subchannel gains
Using Eq. 11, rewrite Eq. 10 as:
In order to obtain the Viterbi decoding metrics dependent on rotated trellis
codeword symbols ,
,
and
,
separately, By combining , ,
and
simplifying Eq. 14, the CIOD group decoding metric can expressed
as:
where, the decoding metrics expressions of rotated trellis codeword symbols:
The Viterbi Algorithm using the branch metrics (Eq. 16) can be easily applied in the ML decoding of the proposed trellis code CIOD.
DESIGN OF TRELLIS CODE
Here, the codeword error probability of the proposed trellis code CIOD will be derived and then jointly optimize the rotation angle θ and trellis code to obtain the best CER performances.
Codeword error probability of proposed trellis code CIOD: Assumed that the receiver have perfectly Channel State Information (CSI), the Pairwise Error Probability (PEP) between the decoder selects erroneous trellis codeword Z and the transmitted codeword X can be expressed as:
where, ,
the received symbols from the vth antenna:
where, and
.
Similarly,
and .
By substituting m(R,X,H) (Eq. 8) the CIOD group decoding metrics
(Eq. 14 and 11) and the corresponding
decoding metrics for m(R,ZH) in Eq. 17. And assuming that
the subchannel noise variables n^{(v)}_{k,i} are i.i.d., zeromean
complex Gaussian distributed with variance 1/2N_{0} per dimension, have:
where,
where ξ = {k,k+1/2K}. In Eq. 18, (d^{2}_{k}+d^{2}_{k+K/2})
is the modified Euclidean distance of a CIOD group pair represented by and
using
the inequality:
upperbound of Eq. 18 can expressed as:
where, the modified Euclidean distance between pair of trellis codewords X and Z.
Let, n_{R} =1 for simplicity, rewrite Eq. 22 as:
The rotated trellis codewords corresponding to X and Z are:
and
respectively. Let, differ
only inside the short part with length κ, i.e., only differs
from .
Equation 23 becomes:
where, η = {s+1,s2,+...,s+κ),
and d(•)t takes the integer operation. The coordinate interleaver π
can be represented by a pair of permutations for real parts Re{π(k)} and
imaginary parts Im{π(k)} of the input vector. According to Eq.
4:
and
Perfect coordinate interleaving means that f(ξ)≠g(ω) for every
pair ξ, ωεη and f(ξ)≠f(ω), g(ξ)≠g(ω),
for every pair ξ, ωεη when ξ≠ω. Hence, after
performing perfect coordinate interleaving, there is no repeated subcarrier
fading coefficients H^{μ}_{k} in Eq. 24.
Assuming that the subcarriers are also perfectly interleaved and transmit antennas
are well separated, hence, the subcarrier fading coefficients H^{μ}_{k}
in Eq. 24 are zeromean i.i.d., complex Gaussian random variables
with variance 1/2 per dimension. Taking the expectation of Eq.
21 over Rayleigh distributed random variable H^{μ}_{k}
using Eq. 24, obtain:
In general, rotation angle θ can be selected such that both of real and
imaginary components of and
do not differ in the case of .
Define two different sets of k values, Re{η} and Im{η} for which real
and imaginary components of rotated trellis codeword symbols differ,
respectively. Assuming in the high SNR region, Eq. 25 can
be expressed as:
The cardinality of sets Re{η} and Im{η} are denoted as Re{η} and Im{η}, respectively, Eq. 26 becomes:
From Eq. 27, under the assumptions of perfect coordinate
and channel interleaving, the achievable diversity of the system for n_{R}
= 1 is:
The maximum value of min Re{η} and Im{η},
is 2Γ, wherein, Γ is the effective code length of the trellis code.
the achievable diversity of trellis code CIOD OFDM is G_{d} = 4Γ,
which is four times of the twosymbol interleaved TCSTBC OFDM (Gong and Letaief, 2003). the trellis code CIOD coding gain is:
The codeword error probability can be expressed in the form of the PEP:
where, P(X) is the probability of the codeword X yielded from the trellis encoder and P(X→z) is the PEP.
Jointly optimization of rotation angle and trellis code: There are two
possible approaches for exhaustive trellis code optimization. The one approach
is to optimize the trellis code and rotation angle θ to maximize the achievable
diversity of trellis code CIOD OFDM system G_{d} (Eq.
28) and the trellis code CIOD coding gain G_{c} (Eq.
29). The other approach is to optimize the trellis code and rotation angle
θ to minimize the upperbound of codeword error probability, which can
be obtained by substituting Eq. 27 in 30. We joint rotation
angle θ and trellis code to settle with minimization problem of the upperbound
of codeword error probability over all possible trellis generator polynomials
using exhaustive search technique as described by Schlegel and Costello (1989).
After performing an exhaustive search minimizing the upperbound of codeword
error probability over all possible trellis codeword pairs X and Z starting
and merging at the common trellis states in the case of {4, 8, 16, 32}state
rate2/3 8PSK trellis code, respectively, the values of rotation angle θ
ranging from 0.5° till 22.5° with step of 2° are selected, wherein,
the high SNR case of E_{s}/N_{0} = 14 dB is considered, the
value of E_{s}/N_{0} is selected to find the optimum trellis
codes at the CER of 10^{2} during the exhaustive search and let κ
= 3, in order to contain the critical codeword pairs having considerable effect
on the CER performance, the value of κ is selected larger for trellises
with larger number of states during the exhaustive search.

Fig. 2: 
Upperbound of codeword error probability vers. rotation angle
with optimal trellis codes 
Table 1: 
Optimum rate2/3 8PSK trellis codes for trellis code CIOD 

From Fig. 2, the upperbound of codeword error probability
with optimal trellis codes for different values of θ are considered for
{4, 8, 16, 32}state trellises. The upperbounds of codeword error probability
for the optimum trellis code decrease with values of rotation angle θ and
achieve their minimum value θ = 22.5°. For 8PSK constellation is used,
rotation angles between 22.5≤θ'≤45°, it result in the same codeword
error probability upperbound values for θ = 45°θ'. The octal
generator polynomials calculated by an exhaustive code search technique for
the optimum trellis codes minimizing Eq. 30 are shown in
Table 1, wherein the optimum rotation angle θ = 22.5°
and the values of achievable diversity gain G_{d} (Eq.
28) and coding gain G_{c} (Eq. 29) are also given.
In Table 1, the 4state trellis code is given by minimizing the upperbound of codeword error probability for E_{s}/N_{0} = 18dB with κ = 4, the 8, 16 and 32state trellis codes are given for E_{s}/N_{0} = 14 dB with κ = 6.
NUMERICAL RESULTS
Assumed that MIMOOFDM channel is static during two consecutive OFDM frames
transmission, the receiver have perfect CSI, perfect random channel interleaving,
OFDM subcarriers K = 128 and a multipath channel taps L = 32 with equal power
assignment.

Fig. 3: 
The CER performances of proposed 4state trellis code CIOD
OFDM with different constellation rotation angles, benchmark twosymbol
interleaved TCSTBC OFDM [GL] and double block type symbol interleaved TCSTBC
OFDM systems 
In Fig. 3, the simulation results shows the CER performances
of the proposed trellis code CIOD OFDM system outperforms that of the twosymbol
interleaved TCSTBC OFDM system by Gong and Letaief, 2003,
wherein, the 4state 8PSK trellis code proposed by Jamali
and LeNgoc (1991).
Both of the systems are configured with two transmit and one receive antennas
and their bandwidth efficiency equal to 2 bits/s/Hz, when trellis code termination
and OFDM cyclic prefix are removed. All of the trellis codewords of length 2K
= 256 were terminated. Figure 3 shows the CER performances
of the proposed trellis code CIOD OFDM system with different values of the rotation
angle θ, the best performance is achieved at θ = 22.5°, it agree
with the trellis code search results given earlier. Nevertheless, the 8PSK
CIOD (Khan and Rajan, 2006) lost its optimality (previously
optimal θ = 4.87°) when trellis code is used. the proposed trellis
code CIOD provides 10 dB SNR gain at the CER of 10^{3} compared to
the twosymbol interleaved TCSTBC (Gong and Letaief, 2003),
the achieved diversity of trellis code CIOD is greater than that of the TCSTBC
from the slopes of the CER curves. When a symbol interleaver is used with TCSTBC,
the set size equal to the effective length of the trellis code. thus, the maximum
achievable diversity of TCSTBC doubles. Figure 3 shows that
2xK block type symbol interleaved TCSTBC outperforms the twosymbol interleaved
TCSTBC (Gong and Letaief, 2003) by 7.5 dB in SNR at
the CER value of 10^{3} and trellis code CIOD outperforms the symbol
interleaved TCSTBC by 2.5 dB in SNR at the CER value of 10^{3}.
In order to evaluate the performance improvement contributed by the trellis
code design as earlier using the trellis code (Jamali and
LeNgoc, 1991) and the trellis code PT4 (Table 1) to simulate
trellis code CIOD OFDM system. Both of the 4states trellises using 8PSK signal
constellation provide bandwidth efficiency of 2 bits/s/Hz. From Fig.
4, the CER performance of trellis code CIOD improves about 0.4 dB SNR when
PT4 is used. Therefore, the proposed optimized trellis code CIOD OFDM offers
approximately 10.7 dB SNR gain at the CER of 10^{3} compared to two
symbol interleaved TCSTBC OFDM (Gong and Letaief, 2003).
The performance of CIOD OFDM (Jagannadha Rao et al.,
2004) is nearly equal to that of the LCPSTBC (Liu
et al., 2002), the CIOD is singlesymbol decodable while LCPSTBC requires
more complex sphere decoding. In Fig. 4, the number of OFDM
subcarriers used with 16state 4PSK SpaceFrequency Trellis Code (SFTC) is
256, hence SFTC has an OFDM codeword with the same number of symbols as the
other systems and Fig. 4 shows that the proposed 4state trellis
code CIOD outperforms the 16state SFTC (Agrawal et al.,
1998) by 6 dB in SNR at the CER of 10^{3}. Figure
4 also shows that the CER performance of the proposed 4state trellis code
CIOD is 2 dB better than that of LCPCIOD (Jagannadha Rao
et al., 2004). The 4state symbol interleaved TCSTBC offers a diversity
gain G_{d} = 4 while offering approximately additional coding gain of
3 dB, but its CER slope is same as CIOD and LCPSTBC.

Fig. 4: 
The CER performance of proposed vers. benchmark OFDM systems 

Fig. 5: 
The CER performance of proposed trellis code CIOD OFDM vers.
benchmark twosymbol interleaved TCSTBC OFDM systems [GL] with {8, 16,
32}state trellis codes 
The performance of STBC OFDM (Lee and Williams, 2000)
shows that the proposed 4state trellis code CIOD OFDM system has superior performance
compared to all of the benchmark systems.
In Fig. 5 and 6, assumed that the interleaved
multipath channel is perfect, the receiver have perfect knowledge of CSI and
OFDM subcarriers K = 256.

Fig. 6: 
The CER performance of proposed {8, 16, 32}state trellis
codes CIOD OFDM vers. the symbol interleaved TCSTBC OFDM systems 
Figure 5 shows the CER performances of {8, 16, 32}state
twosymbol interleaved TCSTBC systems (Gong and Letaief, 2003), wherein, the
trellis codes are denoted by C8, C16 and C32, by Schlegel and Costello (1989)
and the optimized {8, 16, 32}state trellis codes (Table 1)
are used in the proposed trellis code CIOD OFDM. Figure 5
shows that the proposed {8, 16, 32}state trellis codes CIOD OFDM system provides
additional 10.2, 4.8 and 4.2 dB gains in SNR at the CER of 10^{3} compared
to the twosymbol interleaved TCSTBC (Gong and Letaief, 2003).
Figure 6 shows the CER performances of {8, 16, 32}state symbol interleaved TCSTBC, the trellis codes are denoted by C8, C16 and C32, by Schlegel and Costello (1989) and the optimized {8, 16, 32}state trellis codes (Table 1) are used in the proposed trellis code CIOD OFDM. the proposed system provides additional 3, 2 and 1.5 dB gains in SNR at the CER of 10^{3} compared to the symbol interleaved TCSTBC with the {8, 16, 32}state trellis codes, respectively. the CER performances curve of the 8state trellis code CIOD extraordinary agree with that of the 32state symbol interleaved TCSTBC.
CONCLUSIONS
A robust trellis code CIOD OFDM system offering high diversity gain is proposed to considerably boost up the CER performances compared to the benchmark systems. The proposed trellis code CIOD system combines the trellis code, the coordinate interleaver and the STBC to enhance the MIMOOFDM transmission performance. The Viterbi decoding branch metrics of proposed trellis code CIOD is derived from PEP analysis. the optimum {4, 8, 16, 32}state rate2/3 8PSK trellis codes for trellis code CIOD are found by exhaustive search. the simulation results shows that the performances of proposed trellis code CIOD considerably outperforms benchmark systems.
ACKNOWLEDGMENT
Present Research Project was fully sponsored by the Important National Science and Technology Specific Projects of China: Research on mediumhighspeed sensor network system under Grant No. 2009ZX0300600302 conducted from 2009 to 2011, Guangzhou, China.