*2.3.2. Minimum mean-square-error MIMO receiver*

Unlike ZF receiver which completely force the interference to zero, the MIMO MMSE receiver tries to balance between interference mitigation and noise enhancement [39]. Thus, at low SNR values the MMSE outperforms the ZF receiver. In the MMSE MIMO receiver the decoding factor **G** is designed to maximize the expectation criteria of the form:

$$E\{ [\mathbf{Gy} - \mathbf{s}][\mathbf{Gy} - \mathbf{s}]^\* \}\tag{19}$$

By analytically solving this MMSE criterion for MIMO channel, the factor **G** is found to be:

$$\mathbf{G} = (\mathbf{H}^\star \mathbf{H} + \sigma\_n^2 \mathbf{I})^{-1} \mathbf{H}^\star \tag{20}$$

With successive Interference Cancelation (SIC), additional nonlinear steps are added to the original ZF and MMSE equalizers. The resulting versions are ZF-SIC and MMSE-SIC decoding methods. In short, in SIC, the data layer symbols are decoded and subtracted successively from the next received data symbol starting with the highest SINR received signal at each decoding stage. The main drawback of this kind of receive structure is however, the error propagation.

## *2.3.3. Maximum likelihood MIMO receiver*

The Maximum Likelihood (ML) decoder is an optimum receiver that achieves the best BER performance among all other decoding techniques. In ML, the decoder searches for the input vector **s** that minimizes the ML criteria of the form:

$$\left\|\mathbf{y} - \mathbf{H}\mathbf{s}\right\|\_{F}^{2} \tag{21}$$

where . *<sup>F</sup>* 2 denotes the matrix/or vector Frobenius norm. The complexity of this decoder increases exponentially as the number of transmit and receive antennas increases. In spite of its good BER performance, ML decoding is however not used in any practical system.

of the sphere. The correctness of the codeword is in turn dependent on the SNR of the system. The search in Sphere decoding is restricted by drawing a circle around the received signal in a way to encompass a small number of lattice points. This entails a search within sub-set of the codes-words in the constellation and allows only those code-words to be checked. All codewords outside the sphere are not taken into consideration for the decoding operation [8].

On MU-MIMO Precoding Techniques for WiMAX

http://dx.doi.org/10.5772/56034

13

MIMO techniques have been incorporated in all recent wireless standards including IEEE 802.16e, IEEE 802.16m, IEEE 802.11n, and the Long-Term Evaluation (LTE). The WiMAX profile IEEE 802.16e defines three different single user open loop transmission schemes in both

**•** Scheme defined as matrix *A* which describes spatial multiplexing mode of operation for two

**•** Scheme defined as matrix *B* which describes the spatial diversity mode of operation for two different symbol streams through two different antennas with the basic Alamouti Space-

**•** Scheme defined as matrix **C** which combines the respective advantages of diversity and spatial multiplexing modes of operation for two different symbol streams through two

In addition to the basic MIMO techniques supported by the IEEE 802.16e, the profile of IEEE 801.16m also supports several advanced MIMO techniques including more complex configu‐ rations of SU-MIMO and MU-MIMO (spatial Multiplexing and beam-forming) as well as a number of advanced transmit diversity [43, 44]. The profile also defines multi-mode capability to adapt between SU-MIMO and MU-MIMO in a predefined and flexible manner. Further‐ more, flexible receiver decoding mode selection is also supported. Unitary precoding or beamforming with code-book is also defined for both SU-MIMO and MU-MIMO configurations [45]. Cade-book based MU-MIMO precoding techniques found to be effective for the FDD mode of operation because of the great amount of reduction on feedback channel provided, while they are ineffective in the TDD mode of operation [24]. In the next section, we will introduce non-unitary MU-MIMO precoding method to the area of WiMAX. It can be shown that the non-unitary precoding like our proposed method will be applicable and suitable to the TDD mode of operation as accurate CSI is available at the transmitter for the precoding design. In the next section, we will review the most recent researched precoding methods and

Keeping in mind the computational complexity of the nonlinear DPC precoding methods, the research community, as we mentioned before, gives more preference to the investigations of computationally simple linear precoding techniques. Many design metrics and conditions are

*2.3.5. MIMO in the current WiMAX standard*

Time Block Code (STBC) [4].

extend them by proposing our new method.

**3. Linear precoding for MU-MIMO system**

uplink and downlink channel summarized as below:

different symbol streams through two different antennas.

different antennas. More details of these schemes are given in [40-42].

**Figure 3.** Different forms of Multi-element Antennas Channel Configuration
