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To achieve multiple input multiple output (MIMO) in wireless communication, the orthogonal space-time block coding (OSTBC) is evaluated next. At first, the OSTBC design is extended to include Hadamard matrix, referred to in this work, as traditional Hadamard OSTBC. Next, the Hadamard matrix is imposed on the conventional OSTBC, which is referred to, in this work as, Alamouti-Hadamard STBC. Both the traditional Hadamard OSTBC and the conventional STBC are compared with the Alamouti-Hadamard STBC. It will be shown that imposing the Hadamard conditions over the conventional OSTBC, the performance of the OSTBC 2-transmit antenna scheme can be significantly improved in terms of BER performance. All propositions are well supported with analytical derivations.

When signals traverse multipath environment, they are intercepted and then distorted due to non-uniformity of the surfaces on which they impinge. During reception, the signals can be received in error if the channel attenuation is large. The quality of information received in this case will be heavily degraded. Suppose that replicas of this information are transmitted over many different fading channels, then the probability, p, that all the information will fade simultaneously at each time instance, will be reduced considerably [

In the recent time, MIMO technology has improved as a way of achieving a very dependable broadband network in wireless communication systems. For instance, MIMO technology can transfer high data rate depending on the number of transmission branches (diversities). For two transmit diversities, STBC is used [

It was mentioned in [

In this work, the construction of 2-transmit antenna system based on the Hadamard matrix properties is considered first. Then, the Alamoutiorthogonal STBC is extended to include the Hadamard property for improved diversity gain. Again, a consideration will be given to the code construction discussed for higher antenna order in [

We have organised the remaining parts of this paper as follows: system model is discussed in Section 2, with the OSTBC presented in Section 3. Simulation results are discussed in Section 4 and the conclusion in Section 5.

A 2-two transmit diversity system model is involved and will be constructed from the Alamouti code. In Alamouti case, the code was discussed for frequency flat fading channel, h. If the signal of the system to be processed is s, then the received signal after traversing the channel will be of the form:

where z is the noise term of the additive white Gaussian noise (AWGN). h is constructed from the multipath Rayleigh model as:

where u is the gain corresponding to k^{th} path, N is the maximum number of the resolvable multipath, is the phase and N_{T} is the maximum number of transmit antennas. The channel, h, in Equation (1) is a matrix containing, where is the transpose operator. Since the model considers a frequency flat channel, the amplitude for all N-paths of each channel is therefore uniform.

The most pioneering work on STBC was introduced by [_{1} and s_{2} are proper subsets of the encoding matrix s. These can be obtained as [

,. Equation (3) represents a demultiplexing that volunteers equal power sharing with n as the prevailing discrete signal block under consideration.

In [_{1} and s_{2} will be transmitted during the first time slot and the conjugates and will be transmitted in the second time slot. Then the encoding matrix containing the above formulation can be described as:

If signal energy on each antenna branch is normalized to unity, then the matrix s is an orthogonal matrix. Since there are N_{T} transmitting elements, then each of the codes (and their respective conjugates) will be characterized as, where N_{R} is the number of receive elements. For simplicity, N_{R} is limited to 1. The construction above permits that the system attains full transmission rate and full diversity. For instance, there are two time slots, and there exist two antenna spaces over which the transmissions would take place. A system with T time slots and K antenna spaces attains full transmission rate if T/K = 1. In the code structure shown in Equation (4), there exist full transmission rate.

Now, recall Equation (1): The s is the orthogonal STBC from Equation (4), and will be transmitted over two antenna spaces. In that case, the channel over which the signals are transmitted will be, where is the Hermitian transpose. In this case there are a maximum of 2 transmit branches (i.e. N_{T} = 2).

_{1} and s_{2}) is transmitted over antennas T_{1} and T_{2} in the first time slot (ts_{1}), while the other part of the codes (and) is transmitted in the second time slot (ts_{2}).

In the receiver, the decoding can proceed. Meanwhile, for an orthogonal matrix with complex entries, say:

Then, the decoding matrix will be of the form [

or, in an expanded form:

where is an identity matrix for N_{T} = 2 and b is a constant volunteered by multiplexing the input signal over N_{T} antenna branches. So, the decoding matrix for the OSTBC following Equation (6) criteria will become:

where represents a Hermitian transposition operation and. The decoding matrix is a diagonal matrix with no off-diagonal terms. Recall the channel matrix, and Equation (7) such that Equation (1) becomes:

Now, expanding a part of this result:

From Equation (9), it is easy to isolate the equivalent channel matrix of Equation (4) as:

Then, with s_{1} and s_{2} each antenna branch will transmit for the two different time slots:

where. Equation (11) is the case for one receiver. So, using the decoding criteria of Equations (18) and (19) in Equation (1), then:

Equation (12) provides some coding gain according to the amplitude of the channel shown in Equation (2), where:

In Equation (13), the decoding matrix shows the gain due to the channel as. However, the noise term is amplified by the h^{H} term. Wherein there are more than one receive antenna, then each of the h_{1} and h_{2} can be treated respectively as a vector of the forms:

where N_{R} is the maximum number of receivers.

The quasi-cyclic Hadamard matrix orthogonalizes a given code by using the Hadamard matrix of the order of the cyclic matrix. This is the case for higher order matrix. For matrix order of N_{T} = 2, the cyclic matrix can be written as:

Since the Hadamard matrix is a matrix of 1s and -1s, and usually discussed the eigenvectors of an orthogonal matrix, the Hadamard matrix for Equation (15) will be:

Equation (16) is the eigenvector which can orthogonalize Equation (15). Combing Equations (15) and (16), we obtain:

Equation (17) is the encoding matrix to be transmitted. In the receiver, the decoding matrix will be characterized as:

where and. Expanding and, it can be noticed that: and,

In the case, deplete the gain. Also, in case, is also residual.

Notice also that the converse is also true:

Equation (18) can easily be corroborated with the Hadamard criteria identified in [

It can be seen that the decoding matrix revealed by Equation (7) does not provide the b term of Equation (6b). The performance gain of the above technique can be improved by exploiting the Hadamard property of the quasi-cyclic Hadamard codes already discussed in [

Equation (19) can be regarded as the eigenvector of Equation (4) and can be combined as:

Equation (20) is then the encoding matrix of the codes to be transmitted. Notice that Equations (4) and (19) form, independently, orthogonal codes. In the receiver, the decoding matrix will be characterized as:

where,. If the residual terms identified in the case of Equation (18) can be zero, then full diversity can be attained. Only the Alamouti STBC codes satisfy these criteria. Equation (21) provides some processing gain to improve the diversity performance of the system in the order of the number of transmitting elements, in this case, 2. Thus by imposing the Hadamard matrix criteria on the design of orthogonal space time block codes, the system performance can be increased up to the number of transmitter diversity order.

Like in Section 3.1, the equivalent channel matrix (of Equation (20)) can be constructed such as:

From the channel matrix in Equation (10), V_{h} is formed as:

Equation (22) is realized by multiplying the channel matrix of the OSTBC and the equivalent eigenvectors of Equation (23). In that case, the decoding matrix (in the receiver) can be seen as:

The constant 2 can be traced to the theory of orthogonal STBC discussed in [

We demonstrate the realization of the above propositions using some computer simulations performed in a MATLAB^{®} environment. The demonstration is a test of error probability of the combined Alamouti-Hadamard system (depicted as in Figures 2 and 3 “AlamHad”) comparable to the traditional Hadamard STBC (depicted in Figures 2 and 3 as “Traditional”) and conventional STBC (also depicted in Figures 2 and 3 as “Alam”). It involves two system models, namely, BPSK and QPSK. Using the above design conditions, we demonstrate the simulation of the 2-antenna system for BPSK and QPSK mapping schemes.

The simulation is for Rayleigh frequency flat channel model, although each transmission channel branch is independent of another. In one case, the traditional Hadamard STBC model for 2 × 1 system is simulated. The second case involved the conventional Alamouti space-time coding structure for 2 × 1 system. The third case

is a combination of the Alamouti code structure with the Hadamard matrix.

From

In

An approach to improve the design of multi-antenna system for a maximum of 2-antenna diversity branch scheme has been presented and evaluated. It exploited the Hadamard matrix codes in the construction of the 2-transmit antenna system. Two performance scenarios have been shown for BPSK and QPSK systems with a maximum of one-receiver antenna diversity. Results obtained well support the mathematical proposition obtained when the Hadamard matrix is used. This study can then be extended for multiple receiver diversity.