INTRODUCTION
For wireless communication systems, the principal radio design challenges
arise from the harsh radio frequency (RF) propagation environment characterized
by channel fading, due to diffuse and specular multipath and CoChannel
Interference (CCI) due to the aggressive reuse of radio resources. Interleaved
coded modulation on transmit and multiple antennas on receive are standard
methods used by wireless communication system designers to combat timevarying
fading and to mitigate interference. Both are examples of diversity techniques,
which can be provided using temporal, frequency, polarization and spatial
resources. However, the wireless channel is neither significantly timevariant
nor highly frequency selective. Hence the communication engineers consider
the possibility of deploying multiple antennas at both the transmitter
and receiver to achieve spatial diversity and also provide high performance.
The SpaceTime Codes first used in Tarokh et al. (1998), achieve
coding gain, with inputs mapped on the vectors rather than scalars. Teletar
et al. (1995) and Foshini and Gans (1998) had shown independently
that the rich scattering wireless channel can support higher data rates
when multiple antennas are used at the transmitter and the receiver. Alamouti
(1998) gave a simple, single symbol decoding STBC for two transmitter
antennas from the complex orthogonal design matrix. The simple design
rule and an easy decoding technique stimulated a lot of researchers to
work in this area and since then many more STBCs, are proposed. Tarokh
et al. (1999) generalized the Alamouti scheme and gave STBC from
Orthogonal design. Then many more complex Orthogonal design matrices with
rates less than 1 like those in Su and Xiz (2003) were identified. Sethuraman
et al. (2003) proposed STBC from diversion Algebra and extension
field concepts to identify code matrices. Damen et al. (2002) proposed
a diagonal algebraic spacetime code that by construction are of full
rank and of rate 1. But these codes require some parameters to optimize
over for diversity and coding gain. Mukkavilli et al. (2000) used
the equal eigenvalue criterion to maximize the coding gain and identified
some codes using this strategy.
The purpose of this research is to construct a new SpaceTime Block Codes
from Orthogonal Polynomials of full rank, full rate and low complexity.
Simulation results conform that, the proposed spacetime block code with
multiple transmit antennas, a significant performance gain can be achieved
with less processing expense, as the proposed code scheme is configured
as integer only.
STBC DESIGN AND SYSTEM MODEL
STBC design: In Space Time Block Code design, the essential design
criteria are the provided transmit (Tx) diversity, the symbol rate of
the code and the delay. The degree of Tx diversity is characterized by
the number of independently decodable channels. For full diversity it
equals the number of transmit antennas. If multiple receive (Rx) antennas
are deployed the total diversity degree is the product of the Tx and Rx
diversity degrees. The number of Rx antennas is however, irrelevant for
the design of orthogonal STBC. The symbol rate of the code is the number
of symbols transmitted by the code per time. The delay is the length of
STBC frame. Depending on the underlying modulation scheme, the proposed
orthogonal polynomial based STBC is aimed to maximize the rate and minimize
the delay, keeping the full diversity. In general if transmitting antenna
is one and receiving antenna is one then the pairwise error probability
is inversely proposanal to signal to noise ratio (N_{t} = 1, N_{r}=1
then PEP α SNR^{1}). If the transmitting antenna is one
and receiving antennas are more than one the pairwise error probability
is inversely proposanal to signal to noise ratio of power of N_{r}
(N_{t} = 1, N_{r}>1 then PEP α SNR^{–
}Nr) If the transmitting antenna is more than one and receiving antenna
is more than one the pairwise error probability is inversely proposanal
to signal to noise ratio of power of N_{r} and N_{t}.
(N_{t} > 1, N_{r}>1 then PEP α SNR^{Nr
Nt}) and Channel capacity C is defined as min {^{ }N_{t,}
N_{r}}log (SNR). So the rate and diversity is only based on STBC.
General STBC model: We consider singleuser wireless communication
links consisting of N_{t} transmit antennas and N_{r}
receive antenna. The received symbol r_{jk} can be given as,
where, j = 1,2,..,N_{r} denote the receive antenna and k = 1,2,...T
the time at which the symbol was sent, u_{ik} is the code symbol
transmitted from antenna i = 1,2,...,N_{t} at time k and h_{ji
}the complex channel gain between the i^{th} transmit
antenna and the j^{th} receive antenna. The noise symbols
n_{jk} are complex Gaussian with mean zero and variance σ^{2}.
In matrix formulation, this system can be represented as,
where:
H 
= 
The channel matrix of dimension N_{t}xN_{r} 
U (or) M 
= 
The code matrix of size TxN_{t} 
R 
= 
The received matrix of size TxN_{r} 
N 
= 
The noise matrix of size TxN_{r} 
Here M is the spacetime block code matrix. The spacetime block codes
(STBC) spans a matrix of size N_{t}xT, where the i^{th }row
vector is transmitted by the i^{th }transmit antenna and the t^{th
}column vector is transmitted during the t^{th} time slot.
We assume quasistatic fading channels where the channel matrix remains
constant over the code duration T. Perfect channel estimation at the receiver
end is also assumed and the systems have no feedback. Channel estimation
is done with training/pilot sequences in regular intervals during the
transmission. We focus on full diversity designs that have a simple and
effective decoding strategy.
PROPOSED ORTHOGONAL POLYNOMIALS BASED STBC
The proposed space time block coding is considered around a cartezian
coordinate separable, blurring, code operator in which the signal I results
in the super position of point source of impulse weighted by the value
of the object function f. Expressing the object function f in terms of
derivatives of the signal function I relative to the cartezian coordinates
and time is very useful for analyzing the signal in order to achieve the
diversity. Hence, the initial requirements to analyse the diversity may
be stated as follows: Since the diversity can be achieved based on the
local properties of the signal, a local code operator is required to be
devised such that it is cartezian separable and denoising operator. The
two dimensional code function M(x,y) can be considered to be a real valued
function for (x,y) ∈ XxY where X and Y are ordered subsets of real
values. In our case the x is modeled to represent the space and y represents
time slot and consisting of a finite set, which for convenience can be
labeled as {0,1,2...,n1}, the functions M (x, y) reduces to a sequence
of functions
M(i,t) = u_{i}(t), 
I=0, 1, 2, ...n1 
(1) 
As shown in Eq. 2 the process of space time block codes
analysis can be viewed as the linear two dimensional transform coding
defined by code operator, M (x, y) M (I, t) = u_{i} (t)
Considering both X and Y to be finite set of values {0,1,2...,n1} Eq.
2 can be written in matrix notation as follows
where the code operator  M  is
is the outer product  β`_{ij}  and  I 
are the n^{2} matrices arranged in the dictionary sequence. 
I  is the signal to be transmitted and  β`_{ij}
 are the coefficients of transformation.
We consider a set of orthogonal polynomials u_{0}(t), u_{1}(t),...,u_{n1}(t)
of degrees 0,1,2,..., n1, respectively. The generating formula for the
polynomials is as follows.
u_{i+1} (t) = (tμ)
u_{i} (t)b_{i} (n) u_{i1} (t) for i ≥1, 
(5) 
u_{1} (t) = t μ and u_{0} (t) = 1,
where:
and
Considering the range of values of t to be t =i, i = 1,2,3,...,n, we
get
We construct code operators  M  s of different sizes from
the above orthogonal polynomials as follows.
for n ≥ 2 and t_{i} = i+1
Note: For the convenience of code operations, the elements of are scaled
to make them integers.
Construction of code operator: Here, we present the construction
of the code operator of size n. It can be noted at this juncture that
some of the STBC which are proposed in the literature are related with
the code operator of even size only. Also special techniques are devised
to use Hadmard matrix for construction of odd size code operator. But
our proposed orthogonal polynomial based STBC is designed to have any
width. For the sake of computational simplicity, the finite cartezian
coordinate set X,Y are labeled as {1,2,3} to model the space and time
slots respectively. The code operator in Eq. 4 that defines
the linear transform of signals can be obtained as  M 
 M  where M is computed as scaled from Eq. 5
as
The set of 9 two dimensional basis operators O_{ij}, (o≤i,j≤2)
can be computed as follows.
where,
is the (i +1) st column vector of. The complete set of the basis operators
of size (2x2) and (3x3) are given below:
Polynomial basis operators of size (2x2) are
where:
Polynomial basis operators of size (3x3) are
The following symmetric finite differences for estimating partial derivatives
at (x,y) position of the signal I are analogous to the eight finite difference
operators excluding
O_{00}.
and so on
In general.
where,  * indicates the arrangement in dictionary sequence and
(,) indicates the inner product and β_{ij} are the coefficients
of the linear transformations defined as follows.
 β` _{ij} 
= 
 ^{t }  I  
(10) 
where, 
 is the 2D code operator defined as 
 = 


 .
Theorem: The orthogonal transformation (Eq.10) defined by the
orthogonal system 
 is complete.
Proof: We obtain an orthogonal system  H  by normalizing

 as follows
 H  = 
 ( 
 ^{t} 
 ) ^{ ½} 
Consider the following orthonormal transformations
 Z  =  H  ^{t}
 I  = ( 
 ^{t} 
 ) ^{½} 
 ^{t}  I  = ( 
 ^{t} 
 ) ^{½}  β’* 
Since,  H  is unitary,
where,
( 
 ^{t} 
 ) ^{1}  β’* 
(11) 
As per Eq. 11 the signal  I  can be expressed
as a linear combination of the 9 basis operators of which  O_{00}
 is the local averaging operator and the remaining 8 are finite
difference operators (Eq. 9). From Eq.
11, we obtain the completeness relation or Bessel`s equality as follows.
The code word difference matrix ΔM is defined as the difference
between the transmitted code word and the word received by the receiver.
Where ΔM has the same structure as the code matrix M shown above
and explicit evaluation of the determinant shows that the value will be
zero only when the code matrix and received matrix are same. As determinant
is nonzero the code difference matrix is full rank and hence satisfies
the rank criteria.
Maximumlikelihood decoding: Here, we follow the ML decoding as
described by Damen et al. (2003) and the same is detailed below.
In several communication problems, the received signal is given by a linear
combination of the data symbols corrupted by additive noise, where linearity
is defined over the field of real numbers. The inputoutput relation describing
such channels can be put in the form of the real multiple inputmultiple
output (MIMO) linear model
where x ∈R^{m}, y, z, ∈ R^{n} denote the channel
input, output and noise signal and B∈R^{ nxm} is a matrix
representing the channel linear mapping. Typically the noise components
z_{j}, j = 1....n are independent and identically distributed
zeromean Gaussian random variables with a common variance and the information
signal x is uniformly distributed over a discrete and finite set C ⊂
R ^{m}, representing the transmitter code book. Under such conditions
and assuming B perfectly known at the receiver, the optimal detector g:
y →
∈ C that minimizes the average error probability.
is the maximumlikelihood (ML) detector given by
For the sake of simplicity, we assume that C = X^{m}, where X
is a pulse shift key modulation (PSK) signal set (Proakis, 2002) of size
Q,
i.e.,
with Z_{Q} = {0,1,...Q}
Under the assumption (14), by applying a suitable translation and scaling
of the received signal vector (13) takes on the normalized form
where the components of the noise z have a common variance equal to 1.
SIMULATION RESULTS
Here, we present the simulation results for STBC constructed using the
proposed orthogonal polynomial based coding scheme. In our simulation
experiment we use the number of symbols to be 300, with SNR ranging from
10 to 22 dB. With a fixed 3 transmitting antennas and one receiving antenna,
we simulate the proposed orthogonal polynomial based STBC. For the modulation
type PSK, the Bit Error Rate (BER) obtained by the proposed STBC, is plotted
against the SNR range and is shown in the Fig. 1. For
the modulation type QASK, the bit error rate by the proposed STBC, against
the same range of SNR is shown in Fig. 2. In order to
measure the performance of our proposed STBC, we also conduct simulation
experiments with the same inputs on the GSTBC. The output of the GSTBC
is shown in dotted lines in the Fig. 1and 2.
From the outputs, it is evident that the proposed orthogonal polynomial
based STBC is superior than GSTBC.

Fig. 1: 
BER performance comparison of PSK (3Tx and 1 Rx) OPSTB
C = ,
GSTBC = o 

Fig. 2: 
BER performance comparison of QASK (3Tx and 1 Rx) OPSTB
C = ,
GSTBC = o 
CONCLUSIONS
In this research, a new space time block coding based on a set of orthogonal
polynomials that could effectively handle multipathfading has been proposed.
The proposed coding is of full rank, full rate and low complexity. The
proposed coding technique exploits the maximum diversity order for a given
number of transmit and receive antennas. The proposed STBC is also compared
with the GSTBC. Simulation experiments are conducted with PSK and QASK
modulation schemes. At the receiver end the ML decoding is used. It is
interesting to note that the proposed STBC is superior to GSTBC in the
sense of configuration of code operator of any size (odd or even). The
proposed STBC also fully satisfies the code design criteria.