Consider a MIMO system with N_t (N_t = 2) transmit antennas and N_r receive antennas and with quasi-static flat fading of block length T (T = 3). It is assumed that the channel state information (CSI) to be known at the receiver but unknown at the transmitter. The input-output relation of this system can be written as

where the normalization q is to ensure that the (SNR at the receiver) is independent of the number of the transmit antennas (N_t). X is the complex matrix of the transmitted symbols that are drawn constellation. H is the complex matrix that contains all the channel coefficients with zero mean and unit variance. Z is the complex noise matrix, and Y is the complex matrix of the received signals. The entries of Z are assumed to be i.i.d. complex Gaussian random variables with the probability density function (pdf), where stands for the complex Gaussian pdf and for the noise variance.

In this section, the three time slots two transmit antennas STBC schemes has been reviewed. Also, advantageous and disadvantageous of the all schemes are included.

In this subsection, the problem is formulated.

Figure 1 shows a schematic drawing of the proposed STBC scheme. Three data symbols are transmitted consecutively during three time slots. Therefore, full rate is achieved (an STBC rate is defined as the number of

transmitted symbols during time slots). The 1^st antenna transmits the three symbols x₁, x₂,and x₃ during three time slots. The 2^nd antenna transmits three symbols v₁, v₂, and v₃. Now, the three symbols transmitted by the 2^nd antenna will be defined. One possible way to define these symbols is to make vector v (v = [v₁v₂v₃]) orthogonal to vector s (s = [x₁ x₂ x₃]), i.e.. As will be shown in the next subsection this causes the decoding complexity reduces from joint three complex symbols detection to joint three real symbols; moreover results in suitable trace criteria [19] [20] . Then, improve BER performance at high SNR. To achieve this goal, three different STBC schemes are proposed as:

In all of the above schemes, the first row is orthogonal to the second one. However, zero entry in the second row reduces diversity order. The following solution overcomes to this deficiency:

where (i = 1, 2, 3) is the second row in matrix. Therefore, v yields as.

In order to achieve power balance, the symbols in the vector v are transmitted with power. Now, we can present the proposed encoding matrix as:

For full diversity, symbols x₁, x₂, and x₃ are selected from three different symbol constellations. Figure 2 represents these three symbol constellations. From the represented constellations x_i (i = 1, 2, 3) are obtained as [21] :

Power in and guarantees (i = 1, 2, 3), where is the excepted val-

ue of z. and are chosen from the standard QAM constellation. Parameter k is unknown and must be optimized to maximize the MDVs. In the subsection III.B the parameter k will be optimized. Note that despite the first row is orthogonal to the second one is neither orthogonal code nor quasi-orthogonal. Consider

orthogonal code, say, then, in which and is identity matrix.

In the previous subsection, the three different symbol constellations were represented. The parameter k norma-

lizes symbols and. In this subsection the parameter k is optimized.

Where and represent two different codewords and det(.) denotes determinant value of (.).

According to, it is easy to show that is full rank. Note that when, it is possible, although we consider in 3.1. This vouches that determinant is always nonzero. Then,

where (i = 1,2,3), and represents the possible error in symbol.

Lemma 1: has non-vanishing-MDV and for its value is 5.82.

Prove:

Consider M-ary QAM standard constellation, where the real and image components of a symbol can be viewed as M₁-ary standard PAM and M₂-ary standard PAM symbols, respectively (e.g. 8-ary QAM constellation can be considered as 4-ary PAM and 2-ary PAM for real and image components, respectively). Then, for real components, , , and, is the mini-

mum Euclidian distance between the PAM constellation points and here is considered as 1. (i = 1, 2, 3) is

an integer such that.

With the above assumptions, the optimized MDV for 4-QAM (for traceability constellation is assumed 4- QAM) in (11) is achieved for, observe that is compatible with any QAM constellation size. As mentioned before, when, the determinant in (11) is nonzero. To optimize MDV and show non-vanishing-MDV property, define expression D as follows:

where, , and. In fact, D is similar the expression in (11) that normalized by and for traceability is considered. Since the first term in (12) is always nonzero define. Now, suppose two following cases:

case1:, this case yields:

According to and and k, the minimum value is always obtained for and. Then, we define as follows

Note, both and cannot be zero at the same time.

case 2:, this case yields:

, is different from in (13). It is clear for, has minimum determinant value. Then, is defined as follows:

In order to maximize MDVs, equate and:

This equality yields and corresponding MDV is when therefore, the obtained is the maximum for all k.

Note that is assumed, this assumption is compatible with any constellation size.

To illustrate the decoding complexity of the proposed STBC scheme with the ML criterion, the decision metric used for the ML decoder will be derived.

Consider a single antenna at receiver, the ML decoder metric is:

where X, h and y are represented in (1). The objection of the

ML decoder is to obtain optimal X between all of the possibilities which minimize (12). After some manipulations,

where

and

and

From (19), it is clear that is independent of, and and is independent of, and. The minimization of (19) is equivalent to minimize and independently. Therefore, the ML decoding requires a joint detection of three real symbols (O(M^1.5)). Compared with QSTBC scheme in [9] and GSTBC scheme in [12] , the proposed scheme for square symbol constellation has lower decoding complexity than QSTBC scheme and slightly higher decoding complexity than GSTBC scheme. But, for real constellations such as BPSK, the proposed scheme will have highest decoding complexity among the three time slots two transmit antennas schemes.

Simple but important properties of the proposed code are illustrated.

・ full rate and full diversity

It was mentioned that three information symbol are transmitted from two antennas during three time slots. This achieves full rate property. Also, when, the determinant of the difference matrix is always nonzero. This ensures full diversity property.

・ Non-vanishing MDVs

It was proved in lemma 1 always has nonzero determinant that guarantees achieves full diversity, and its MDVs do not vanish by increasing symbol constellation sizes. Non-vanishing MDVs property distinguishes our scheme from other schemes have been proposed for LTE-A systems.

・ Compatible with single transmit antenna

Our scheme has the property that first row is in (8), such property provide backward compatible for single antenna, which is desired in LTE-A. QSTBC encoding matrix in (3) is also compatible with single antenna system while has not such property.

We first give performance comparison between and other 3-time 2-antenna STBC schemes. Figure 3 shows the simulation results for the proposed STBC scheme for two transmit antennas and one receive antenna and compares BER performance with HSTBC in [7] , QSTBC in [8] [9] , and GSTBC in [12] scheme. Also, Alamouti scheme (two symbols are transmitted during three time slots) is considered. The transmitted symbols are 4-QAM modulated for all of the schemes except Alamouti scheme (8-QAM modulated), hence their spectral efficiencies are 2 bpcu. As can be observed from Figure 3 the proposed STBC, QSTBC, GSTBC, and Alamouti schemes have the same diversity order, but hybrid STBC scheme does not achieve full diversity. Also, from Figure 3 it is clear that even with lower decoding complexity, the proposed scheme outperforms the QSTBC scheme. However, the proposed scheme with higher decoding complexity than GSTBC scheme has same similar performance at high SNR. For 4-QAM modulation, Note that the MDVs for the proposed scheme, QSTBC, and GSTBC are 5.82, 7.18, and 16, respectively. Therefore, it is expected that the both QSTBC and GSTBC schemes to have better BER performance upon the which is in contrast with simulation results. But, beside rank (diversity) and determinant criteria, trace criterion must be considered, too. The trace criterion state:

In uncorrelated Rayleigh fading, the lowest expected value for the union bound to the pairwise error event is obtained when for all pairs (transmitted matrix) and (detected matrix) the matrix is diagonal with all diagonal elements equal. Alternatively, the best suboptimal codes are those for which the matrices are such that the main diagonal elements are as close as possible to each other, and the row-wise sum of the absolute values of the elements off the main diagonal is as small as possible for each row.

The matrix for the proposed scheme is diagonal with unequal main diagonal elements. However, both QSTBC and GSTBC schemes suffer from nonzero off the main diagonal elements that increases the absolute value of the off the main diagonal to main diagonal ratio. According to the trace criterion the proposed scheme has better BER performance upon QSTBC and GSTBC schemes with equal MVDs. In contrast, according to the

determinant criterion, the proposed scheme should have worse BER performance than QSTBC and GSTBC scheme. Therefore, there is a tradeoff between lower determinant criterion and good trace criterion. This tradeoff closes BER performance for all schemes at high SNR for 4-QAM modulation. Figure 3 shows BER performance of the proposed STBC scheme and the QSTBC scheme for 16-QAM modulation (4 bpcu).

It is clear that the QSTBC scheme because of nonzero values off the main diagonal and lower MDV (for 16- QAM its MDV is 0.12) has poor BER performance compared with other schemes, e.g. at BER 10⁻⁴ both the GSTBC and proposed scheme about 1.5 dB work better than the QSTBC scheme. The MDVs of the proposed scheme and GSTBC scheme for 16-QAM modulation are 16 and 5.82, respectively. Nevertheless, like 4-QAM modulation both schemes have same BER performance at high SNR.

In this subsection, BER curve for with 4 bps throughput is presented and compared with other schemes in Figure 4. However, 64-QAM modulation is used for and 16-QAM modulation for other schemes simulation results show the second proposed scheme has lower BER than other schemes. Because the diagonal elements of the matrix in the second scheme are closer to each other can achieve trace criterion better than the first scheme. In other side, has higher coding gain than that result in good determinant criterion. Therefore, both trace and determinant criteria are dominant compared with other schemes. The profit of this can be seen in Figure 4 that our second proposed scheme outperforms 0.3 dB in power efficiency.