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Recently, space time block codes (STBCs) are proposed for multi-input and multi-output (MIMO) antenna systems. Designing an STBC with both low decoding complexity and non-vanishing property for the Long Term Evolution Advanced (LTE-A) remains an open issue. In this paper, first our previously proposed STBC’s non-vanishing property will be completely described. The proposed STBC scheme has some interesting properties: 1) the scheme can achieve full rate and full diversity; 2) its maximum likelihood (ML) decoding requires a joint detection of three real symbols; 3) the minimum determinant values (MDVs) do not vanish by increasing signal constellation sizes; 4) compatible with the single antenna transmission mode. The sentence has been dropped. Second, in order to improve BER performance, we propose a variant of proposed STBC. This scheme further decreases the detection complexity with a rate reduction of 33%; moreover, non-vanishing MDVs property is preserved. The simulation results show the second proposed STBC has better BER performance compared with other schemes.

Space-time block codes (STBCs) are known as well-suited techniques that provide an effective diversity method to mitigate the fading in wireless channels. In these codes, transmitted signals are repeated in different time slots by using two or more transmit antennas. In order to provide diversity gain, each replica of a signal must encounter independent fading. Thus, transmit (receive) antennas must be separated appropriately. Therefore, if each replica of transmitted signals encounters independent fading, the probability of occurring deep fading is very unlikely. Alamouti code is the most popular STBC scheme for two-transmit antennas systems [

The Three Generation Partnership Project (3GPP) started the next generation wireless systems (4G) under the project Long Term Evolution Advanced (LTE-A) in 2008 [^{2})), where M is size of the used symbol constellation. The second problem is that the minimum determinant values (MDVs) extremely vanish by increasing the symbol constellation size. Recently, Fast-Group-Decodable STBC (Fast-GSTBC) scheme has been proposed in [^{1})) [^{1})). However, GSTBC scheme reduces decoding complexity for 3-time slots two transmit antennas its MVDs vanish. Also, this scheme is not compatible with single antenna.

In [^{1.5}) is proposed, 2) a 3-time slots code rate 2/3 STBC scheme which can achieve full diversity with symbol-wise decoding complexity is obtained. Also, we will show that both schemes have non-vanishing MDVs property. The simulation results show that our first scheme has the same bit error rate (BER) performance with the GSTBC scheme. However, BER for the second proposed scheme is improved about 0.3 dB compared with the first scheme.

The rest of the paper is organized as follows: Section 2 comprises two subsections: 2.1. Channel Model, 2.2. Code Design Criteria, and 2.3. Review of Three Time Slots Two Transmit Antennas STBC Schemes. In Section 3, the Non-Vanishing MDVs Code Rate One 3-Time Slots STBC is introduced. This section includes four subsections: 3.1. Encoding matrix, 3.2. Parameter k Optimization, 3.3. Decoding Complexity, and 3.4. Some

Notations: Hereafter, ^{H}, A^{T}, and tr(A) denote the conjugate-transpose, transpose, and trace of A, respectively;

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 _{t}). X is the

Recently STBC schemes mainly rely on analysis of the pair wise error probability (PEP)

the probability that

the difference matrix

minates the steepness of the Bit Error Rate (BER) curve. Thus, in STBC scheme design ensuring full diversity is important at high SNR values. Afterwards, coding gain should be maximized for given average transmit power that leads to a good determinant criterion. The maximum coding gain results in the minimum PEP. Besides maximizing coding gain, this value should be constant for any symbol constellation sizes. This property is called non-vanishing MDV and has been established for several popular STBC schemes [

The trace criterion is less known but paramount for designing non-orthogonal STBC schemes [

genvalues of the

and for which the row-wise sum of the absolute values of the elements off the main diagonal is as small as

possible. Moreover, the

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.

The hybrid scheme,

However, the scheme achieves code rate one and its decoding complexity is linear at receiver does not achieve full diversity.

Incapacitation of Hybrid STBC to achieve full diversity was good reason for author in [

where

^{2}). There is two defects with

Recently Fast-Group-Decodable STBC (Fast-GSTBC) scheme has been proposed in [

So, design of 3-time slots two-transmit antennas STBC scheme with non-vanishing MDVs is required. In the next section a novel STBC structure with non-vanishing MDVs property that has been proposed in [

In this section, initially the encoding matrix is presented. Then, parameter k is optimized to maximize the MDVs. Also, we will prove that our scheme achieves non-vanishing MDVs. Finally, the decoding complexity of the proposed STBC scheme with ML criterion is illustrated.

In this subsection, the problem is formulated.

transmitted symbols during time slots). The 1^{st} antenna transmits the three symbols x_{1}, x_{2},and x_{3} during three time slots. The 2^{nd} antenna transmits three symbols v_{1}, v_{2}, and v_{3}. 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 = [v_{1}v_{2}v_{3}]) orthogonal to vector s (s = [x_{1} x_{2} x_{3}]), i.e.

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

In order to achieve power balance, the symbols in the vector v are transmitted with power

For full diversity, symbols x_{1}, x_{2}, and x_{3} are selected from three different symbol constellations. _{i} (i = 1, 2, 3) are obtained as [

Power

ue of z.

orthogonal code, say

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

lizes symbols

Where

According to

where

Lemma 1:

Prove:

Consider M-ary QAM standard constellation, where the real and image components of a symbol can be viewed as M_{1}-ary standard PAM and M_{2}-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,

mum Euclidian distance between the PAM constellation points and here is considered as 1.

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

where

case1:

According to

Note, both

case 2:

In order to maximize MDVs, equate

This equality yields

Note that

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

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 ^{1.5})). Compared with QSTBC scheme in [

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

・ Non-vanishing MDVs

It was proved in lemma 1

・ Compatible with single transmit antenna

Our scheme has the property that first row is

It was shown

The structure of

At the above expression, the last term is called symbol interference term. We can decrease interference by omitting one of the information symbols, say

and

where

Lemma 2: _{ }

The determinant for

Like lemma 1 Consider M-ary QAM standard constellation, where the real and image components of a symbol can be viewed as M_{1}-ary standard PAM and M_{2}-ary standard PAM symbols, respectively.

is always greater than zero, we define

Then,

where

where m_{i} (i = 1, 2) is an integer such that

Assume following two cases:

Case 1:

This case results in:

When

Case 2:

This case results in:

・ Compared with

and

since

In general,

In this section, the simulation results of the proposed schemes,

We first give performance comparison between

In uncorrelated Rayleigh fading, the lowest expected value for the union bound to the pairwise error event is obtained when for all pairs

The matrix

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.

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

Diversity Order | Compatible with LTE-A | Compatible with Signal Antenna | Non-vanishing MDVs | Detection Complexity | Code Rate | |
---|---|---|---|---|---|---|

2 | No | Yes | Yes | 1 (Linear Decoder) | 0.67 | |

1 | Yes | Yes | No | 1 (Linear Decoder) | 1 | |

2 | Yes | Yes | No | 2 (ML) | 1 | |

2 | Yes | No | No | 1 (ML) | 1 | |

2 | Yes | Yes | Yes | 1.5 (ML) | 1 | |

2 | Yes | No | Yes | 1 (ML) | 0.67 |

In this paper, a novel STBC structure for three time slots and two transmit antennas was proposed. Based on this structure, two STBC schemes were proposed. The first scheme achieves full rate and full diversity properties and has a joint three real symbols decoding complexity (O(M^{1.5})). Also, the minimum determinant value is constant for different symbol constellation sizes. Then, the proposed scheme achieves non-vanishing-MDV property. Also, the proposed scheme has the property that first row is