## Introduction

Compared to a conventional single antenna system, the channel capacity of a multiple antenna system with *N _{T}* transmitters and

*N*receiver antennas can be increased by a factor of min(

_{R}*N*,

_{T}*N*), without using additional transmission power or spectral bandwidth. Due to the ever-increasing demand of faster data transmission speeds in recent or future telecommunication systems, multiple antenna systems have been actively investigated and successfully deployed in the emerging broadband wireless access networks (for example, Mobile WiMAX)

_{R}MIMO technologies can be divided into two main categories: diversity and spatial-multiplexing. The diversity techniques intend to receive the same information-bearing signals in multiple antennas or to transmit them from multiple antennas, thereby improving the transmission reliability. In the spatial-multiplexing techniques, on the other hand, the multiple independent data streams are simultaneously transmitted by the multiple transmission antennas, thereby achieving a higher transmission speed.

## Diversity and space-time coding

Diversity techniques are based on the assumption that the probability that multiple statistically independent fading channels simultaneously experience deep fading is very low. There are various ways of realizing diversity gain, including the following ones:

- Spatial diversity — sufficiently separated multiple antennas are used to implement independent wireless channels.
- Polarization diversity — independent channels are implemented using the fact that vertically and horizontally polarized paths are independent.
- Time diversity – the same information is repeatedly transmitted at sufficiently separated time instances.
- Frequency diversity — the same information is repeatedly transmitted at sufficiently separated frequency bands.
- Angle diversity — multiple receiver antennas with different directivity are used to receive the same information-bearing signal at different angles.

Time, frequency and spatial diversity techniques are illustrated in Figure 1 below. In time diversity, data is transmitted over multiple time slots. In frequency diversity, the same data is transmitted at multiple spectral bands to achieve diversity gain and in spatial diversity, data is transmitted using multiple antennas:

**Figure 1: Various diversity techniques**

## Alamouti block code

In Nutaq’s OFDM reference design, the diversity technique used is called Alamouti block code. It is a complex space-time diversity technique that can be used in 2×1 MISO mode or in a 2×2 MIMO mode. The Alamouti block code is the only complex block code that has a data rate of *1* while achieving maximum diversity gain. Such performance is achieved using the following space-time block code:

**Figure 2: Alamouti space-time diversity technique**

Briefly, two antennas are used, to send two OFDM symbols and their conjugate, in two time slots, which brings a diversity gain without having to compromise on the data rate. Over the air, the transmitted symbols will suffer from channel fading and at the receiver, their sum will be received. Here is the schematic diagram of an Alamouti wireless system in 2×2 MIMO mode:

**Figure 3: A 2×2 MIMO wireless system using the Alamouti block code**

Since the transmission is done over two periods of time, the decoding will also be done over two periods of time. At the receiver, the received vector ** Y** can be represented by the following equation:

This is for the first time period. For the second time period, the equation is as follows:

where represents the received OFDM symbol at the first time period, for antennas 1 and 2, respectively, and where represents the received OFDM symbol at the second time period for antennas 1 and 2, respectively. Both equations can easily be combined and arranged to produce the following result:

The next step is to find a way to isolate the transmitted symbols, *x _{1} *and

*x*. One way to reduce the number of unknowns is by using a channel estimator to estimate the channel coefficients. In Nutaq’s OFDM reference design, channel estimation OFDM symbols are sent with each transmitted packet to enable estimating those channel coefficients at the receiver. Given the following matrix:

_{2}we can isolate *x _{1} *and

*x*by simply multiplying the matrix

_{2}**by the inverse of**

*Y***. However, since this matrix is not square, we need to use the Moore-Penrose pseudo-inverse**

*H***to solve our equations:**

*H*^{+}Using this inverse matrix expression, the noisy estimated transmitted symbols can be found using the following expression:

The last step would be to make a final decision on the transmitted symbols. In Nutaq’s OFDM reference design, the decision is made based on the minimum squared Euclidian distance criterion. In the next figure, we can see that the addition of diversity to the system brings a significant performance gain in terms of BER in simulation:

**Figure 4: BER curves of different MIMO schemes compared to a SISO implementation**

## Conclusion

The Alamouti space-time block coding is a simple MIMO technique that can be used to reduce the BER of a system, at a specific SNR, without any loss on the data rate. The presented decoding technique is called *hard decision-based zero forcing* and is probably the simplest to implement in hardware. It is a linear decoding technique that has a low complexity. That being said, further reductions in BER can be achieved by using non-linear decoding techniques, which are usually better than linear techniques, and also by using soft decision techniques, with specific QAM constellation decision boundaries.

## References

- Cho, Yong Soo, Jaek won Kim, Won Young Yang, and Chung G. Kang. 2010. “MIMO: Channel Capacity.”In
*MIMO-OFDM Wireless Communications with MATLAB®, 263-280.*Singapore: John Wiley & Sons (Asia) Pte Ltd. - Alamouti, S. M. 1998. “A simple transmit diversity scheme for wireless communications.”
*IEEE J. Select. Areas Commun.*16(8):1451-1458.