Spatial Diversity

2.2.1.1 Receive Antenna Diversity

Receive antenna diversity is a classic low-complexity diversity technique, where multiple antennas are employed at the receiver. The independent signal paths received at the receive antenna are combined before recovering the transmit signal. In order to allow the received multiple-antenna-signal to be independent, the separation among the receive antennas must be sufficiently high. This condition makes the receive diversity techniques suitable for the Uplink (UL) where the base station is capable of accommodating multiple receive antennas.

Another factor affecting the system’s performance is the choice of diversity combining techniques, each of which treats the treatment of the combined signals’ phase and amplitude differently at the receiver. The receive diversity combining techniques are often divided into four main categories [66,67], as follows

• Selection Combining (SC): The block diagram of the SC scheme [54], where n_Rantennas are employed at the receiver, is shown in Fig. 2.1, where the signal having the highest instantaneous SNR out of n_R received signals is selected at every symbol interval as the output, implying the detection of the best incoming signal. In reality, the signal with the highest power at the receiver is often selected, since it is hard to estimate the SNR. Naturally, this combining scheme does not perform well in the low SNR region, because it is likely to select the signal contaminated by the highest noise power. This technique does not require CSI. Hence, it can be employed in both coherent and non-coherent modulation schemes.

• Switched Combining (SwC): In the switched combining scheme [68], the receiver scans all

2.2.1. Spatial Diversity 26

Logic Selector

Detector Rx1

Rx2

RxnR

Figure 2.1: Selection combining.

signal paths received at the nR antennas and chooses a particular path, which has an SNR higher than a predefined SNR threshold. The chosen path is detected and remained, until its signal’s SNR becomes lower than the threshold. In this case, the receiver will scan the received multipath signal again and will switch to another path, where the signal power is higher than the threshold value. In contrast to the SC scheme, which scans all received signal paths continuously and selects the path with the highest SNR value, the SwC scheme only detected the selected path upon its SNR still higher than the SNR threshold. Thus, it can reduce the system’s complexity at the cost of an inferior performance, since the signal associated with the highest SNR might be ignored. Similar to the SC, the switched combining scheme can be employed in conjunction with both coherent and non-coherent modulation schemes.

• Equal Gain Combining (EGC): The EGC technique [54] does not require the knowledge of the fading amplitude at the receiver. Thus, the weighting factor wi is simplified to [66]

wi = e^−jφi, (2.1)

where φi is the phase of the channel coefficients. Compared to MRC, the EGC’s performance is only slightly inferior, but its complexity is significantly reduced, since the knowledge of the fading amplitudes is not required. This technique is known as suboptimal. For coherent detection, the EGC can only be applied in conjunction with modulation schemes having identical symbol energies. If coherent detection is unavailable, the EGC must rely on non-coherent detection techniques such as Frequency Shift Keying (FSK) or DPSK modulation.

• Maximum Ratio Combining (MRC): When accurate CSI is available at the receiver, the most potent MRC linear combining method [54] may be employed. As shown in Fig. 2.2, the signals received from nR antennas are individually weighted, before being phase-coherently combined. The weighting factor wi (i = 1, 2, ..., nR) may be chosen as [66]

wi= h^∗_i

n_R

P

i=1

h^∗_i · h_i

, (2.2)

where hiis the channel coefficient at branch i. This combining technique is capable of maximizing the output SNR [66]. Thus it is known as the optimum receive diversity combining method.

Due to the requirement of accurate CSI, the technique is only applicable to systems employing coherent detections.

2.2.1. Spatial Diversity 27

Adder Detector w₁

w2

wnR

Rx₁

Rx₂

RxnR

Figure 2.2: Maximum ratio combining, where the weighting factors are inserted before the adder, which replaces the logic selector in Fig. 2.1.

2.2.1.2 Transmit Antenna Diversity

Exploiting multiple transmit antennas is another diversity technique, which requires the signals to be preprocessed or precoded before being allocated to the transmit antennas for transmission. Due to its substantial space requirements and signal preprocessing, the arrangement is more suitable for Downlink (DL) scenarios.

Consider a MIMO system equipped with nT transmit and nR receive antennas. Assuming that the data stream is encoded into a ST codeword of size (n_T × T ), where T is the ST block length, the signal vector received in time-slot t (t = 1, 2, ..., T ) is given by

y[t] = Hs[t] + n[t], (2.3)

where H represents the (nR× nT)-element channel matrix, while n is the AWGN vector associated with a covariance matrix of N0InR. All the T vectors may be stacked together, yielding

Y = HS + N , (2.4)

where Y = [y[1] y[2] · · · y[T ]] and N = [n[1] n[2] · · · n[T ]] are matrices of size (nR× T ).

When the CSI is available and the ML detector is employed at the receiver, the estimated codeword may be obtained as

S = arg minˆ S

¯¯¯

¯Y − HS¯

¯¯

¯²= arg min S

XT t=1

¯¯¯

¯y[t] − Hs[t]¯

¯¯

¯². (2.5)

If the receiver detects a codeword other than a transmitted codeword, an error will occur.

a. Space-Time Block Codes

One of the renowned transmit diversity schemes is constituting by STBC, which was conceived by Alamouti [47] for a two-transmit antenna aided system. This scheme was then further developed by Tarokh et al. [48] for multiple transmit antennas.

An STBC may be represented by an (nT × T )-element encoding matrix, where each column rep-resents a time slot and each row reprep-resents a specific antenna’s transmission time slot. For example,

2.2.1. Spatial Diversity 28 Alamouti’s coding matrix for a two-transmit antenna aided system may be formulated as

S =

¯¯

¯¯

¯

s₁ −s^∗₂ s₂ s^∗₁

¯¯

¯¯

¯,

where^∗ denotes the complex conjugate. In the first transmission time slot, the symbols s₁ and s₂ are transmitted by antenna 1 and antenna 2, respectively. In the next slot, the signal -s^∗₂ is sent from antenna 1, while the signal s^∗₁ is from antenna 2.

In order to achieve the maximum attainable diversity gain, the STBC has to satisfy the so-called orthogonal design criterion, which was derived by Tarokh et al. in [48]. Briefly, the encoding matrices have to satisfy the orthogonal property, which may be mathematically represented as

S.S^H = c(|s₁|²+ |s₂|²+ · · · + |s_nT|²)I_nT, (2.6) where c is a constant, while I_nT represents an (n_T × n_T) identity matrix.

Each STBC is also represented by a code rate of R = k

T, (2.7)

where k is the number of symbols in each encoding block while T is the number of time slots used.

The studies of [59,69,70] showed that only Alamouti’s orthogonal STBC design is capable of achieving the maximum attainable rate of unity, which is also often termed as having full-rate. By contrast, other space-time codes have to sacrifice some proportion of their data rate for the sake of maximizing the diversity gain.

In contrast to the family of orthogonal STBC codes, which suffers throughput loss, the class of quasi-orthogonal STBCs and their design criteria was proposed by Jafarkhani [64]. The design allows a higher achievable code rate at the cost of imposing Inter-Symbol Interference (ISI) owing to the non-orthogonal transmissions from the different antennas/time-slots, which degrades the BER performance.

b. Space-Time Trellis Codes

STTCs were first proposed by Tarokh et al. [48, 60], which constitute further development of the conventional trellis codes [71] for multiple antenna aided systems. These codes are designed to achieve both a diversity gain and a coding gain. The design criteria conceived for PSK modulation are detailed in [72], while the general design criteria are described in [73]. Similar to classic trellis codes, each STTC is described by a trellis, where the number of nodes in the trellis diagram corresponds to the number of encoder states. Each node has a specific number of groups of symbols, which is known as the constellation size while each group consists of n_T entries corresponding to the symbols transmitted from n_T transmit antennas.

For instance, the trellis diagram of a 4-QAM scheme associated with a four-state trellis code and two-transmit antennas is illustrated in Fig. 2.3. The trellis has four nodes corresponding to four states.

Each node has four legitimate entries constituted by two symbols, which are the outputs assigned to the two transmit antennas. The outputs {0, 1, 2, 3} are mapped to the 4-QAM modulated symbols {1, j, −1, −j}. More particularly, observe in Fig. 2.3 that assuming the current symbols are 03 and the current state is 0, the output of 3 and 0 are assigned to antenna 1 and antenna 2, respectively,