The reference scenario is shown in Figure 1 and is based on a base station (BS) with transmit antennas, cooperative relay stations (RSs), each one with only one antenna for transmission and reception, and user's terminals (UT), also with one receive antenna each. We assume that the users cannot be reached by the BS directly. The strategy used is the *Decode-and-Forward* in a half-duplex transmission; that is, in phase I, the BS transmits and RSs receive first link/hop and, in phase II, the relays transmit and UTs receive second link/hop. The system uses subcarriers that can be allocated to different users in an OFDMA transmission; that is, different UTs use disjoint sets of orthogonal subcarriers. We assume, for simplicity and without loss of generality, that the subcarriers used in the link BS-RS are the same as in the link RS-UT. The algorithm or policy for the scheduler to assign subcarriers is beyond the scope of the paper. We will consider the transmission of OFDMA symbols as a block and denote a packet as a group of several blocks. In general, can take any value. However, for the space-time block code-(STBC-) based schemes that we are proposing, the block size must necessarily equal the number of transmit antennas, that is, . This is because we are proposing the use of full-rate STBC.

The frequency-domain transmitted signal from the BS is

where is the signal transmitted from the antennas at th subcarrier during block of OFDMA symbols, is a generic precoding matrix , and are the complex base band data to be sent on the th subcarrier by all the transmit antennas, assumed here to be -QAM or -PSK modulated without loss of generality.

Next, the frequency-domain received signal at the th relay on the th subcarrier after discrete fourier transform (DFT) and discarding the cyclic prefix (CP) can be written as

where is the received signal by relay at subcarrier , is the channel frequency response for relay at subcarrier from all the transmit antennas (), and is the zero-mean additive white Gaussian noise (AWGN) vector, with each component () with variance . We can arrange the signal received by all the relays in a matrix form as

where is the received signal by all the relays at th subcarrier during a block of OFDMA symbols, the matrix accounts for the channel frequency response on th subcarrier, and contains the zero-mean AWGN. The th subcarrier can be assigned to any user by the scheduler.

For the second hop, namely, from RS to UT, the frequency-domain joint transmitted signal (It should be noted that each relay transmits one of the rows of the joint matrix . Thus, the precoding matrix must be diagonal, otherwise relays would have to share transmission information, and therefore the complexity would increase, which is not the case) is

where is the signal transmitted by relays at th subcarrier during the block of OFDMA symbols, is a new generic precoding matrix for the second hop, and is the estimated from received and the remodulated transmitted signal. Since the relays are equipped with only one antenna, the estimated signal is performed in a multiple-input single-output (MISO) way by each relay. In this paper, a simple zero-forcing (ZF) equalization and detection is used for reducing complexity at relays and user's terminals. This yields the following frequency-domain received signal at user's terminal

where is the received signal for user at th subcarrier during the block of OFDM symbols, is the channel frequency response for user from the relays at th subcarrier, and is a second AWGN noise vector for subcarrier with each component of variance . Again, grouping all the received signals by users into a matrix yields

being the received signal by all the users on subcarrier during the block of , the matrix the channel frequency response from relays to users at th subcarrier, and a second AWGN matrix. Note that since the system uses OFDMA, at reception, each UT selects the subcarriers with data allocated to it among all the received subcarriers.

In this paper, the evaluation of the performance is based on the bit error rate (BER) as a measurement over different Signal-to-noise ratios (SNR). In the scenarios, there are two different links, one from BS to RS and another from RS to UT. Thus, we define the SNR for each link separately. In addition, since the system is MIMO-OFDMA-based, there will exist different channels (in the first link) over different subcarriers. For these reasons, the average SNR per link is defined as

Looking at (7), the SNR is calculated, averaging the signal over the transmit antennas and the subcarriers. In this way, a single value per link is obtained to associate with the performance in a given scenario. When transmitting from relays, we will have different channels, and in (7), should be replaced by the number of transmitting relays for the scheme () and by .

It should be noted here that the SNR is used as a way of describing different scenarios for evaluation purposes, but it is not a parameter that needs to be estimated to perform the transmission.

### 2.1. A Non-CSI-T Scheme: 2-Hop Space-Time Block Code (2h-STBC)

Although Virtual Maximum Ratio Transmission (VMRT) does not need CSI-T at the relays because the UTs compute the beamforming weights (see Section 3), the selected terminal (and only the selected one) must send its weights to the relays regularly. For this reason, in order to compare and evaluate the impact of channel estimation errors and the use of LDPC codes of the proposed VMRT with the case where no CSI-T is needed, a 2-hop space-time block code is used, denoted as *2h-STBC* throughout the paper; this encoding scheme uses STBC codes in both links. In phase I the BS transmits using Alamouti [27] when using 2 antennas or when using 4 or 8 antennas, [28, 29] which is denoted as *"Alamoutitation"* in [29]. For this scheme, the precoding matrix in (1) is and the number of OFDMA symbols per block () is set to . Thus, the transmitted signal can be written as

with when , or 8, respectively, and

being the matrices containing the data to be sent. are the data on subcarrier at OFDMA symbol .

All the relays will receive the signal, and thus they are able to decode it, *that is*, in (2). Grouping all the received signals by all relays, (3) yields

Therefore, a cooperative *Virtual STBC* transmission can be carried out from RS in phase II, assuming that the RSs are numbered and perfectly synchronized. Now, each relay, or a group of relays, acts as an antenna re-encoding the received signal into . Again, in the general expression of (4), the pre-coding matrix is , and thus arranging all the transmitted signals from the relays into a matrix form, we obtain

with for , or 8, respectively, and

with being the re-encoded signal transmitted by the RS at th OFDMA symbol (). Some observations must be pointed out here. The first one is that a different number of transmit elements can be used on each link; that is, can be different from and ; in fact, usually . Since all the relays decode the transmitted signal by BS, the increase in the number of *virtual* transmitters (relays) will exploit diversity and array gains, and the second one is that the transmitted information by relays may not be orthogonal anymore because each relay decodes the received data and some errors can appear. Thus, some degradation in the performance can be expected at the user's end, especially for the channel estimation algorithm and/or LDPC codes. This scheme is the simplest method to obtain diversity from both links, so we will use it as a reference. Moreover, it can be noted that no CSI-T is needed, but rather only channel state information at the receiver (CSI-R) for coherent demodulation, at both links.