Spatial Diversity

One technique to exploit diversity that may not suffer from the time delay (or equivalently, bandwidth deficiency) which is unavoidable in the system using time diversity technique is antenna diversity or space diversity. The technique is realized by placing multiple antennas at the transmitter and/or the receiver [R2, R24]. The multiple antennas are physically separated by a sufficient distance to make the in- dividual received signals uncorrelated. The separation requirement depends on the frequency, the antenna height as well as the propagation environment. To achieve the uncorrelation property, an antenna separation of a few wavelengths is usually enough.

Depending on whether multiple antennas are used for reception or transmission or both, space diversity can be classified into three categories (see Fig. 2.8): receive diversity (single-input multiple-output, SIMO, channel), transmit diversity (multiple- input single-output, MISO, channel) or transmit and receive diversity (multiple-input multiple-output, MIMO, channel). In the receive diversity, multiple antennas are used at the receiver to gather independent replicas of the transmitted signals without an increase in the transmit signal power or bandwidth. In the transmit diversity, multiple antennas are employed at the transmitter. The source messages are precoded at the transmitter and then spread across multiple antennas. The last type of diversity is the combined version of the transmit diversity and the receive diversity, in which multiple antennas are employed at both the transmitter and receiver sides.

(a) (b) (c) Transmitter Receiver 1 Receiver 2 Transmitter 1 Transmitter 2 Receiver Transmitter 1 Transmitter 2 Receiver 1 Receiver 2

Receive Diversity

In receive diversity, the independent paths associated with multiple receive an- tennas are combined at the receiver. For example, consider a flat fading system with 1 transmit antenna and K receive antennas. Given the transmitted signal s[n] with average symbol energy of Es, the K received signals are

yk[n] = ak[n]s[n] + zk[n], k = 1, . . . , K, (2.17)

where ak[n] represents the fading channel coefficient from the transmitter to the

kth receive antenna at the receiver and zk[n] models AWGN with one-sided power

spectral density of N0. Let y[n] = [y1[n], . . . , yK[n]]T, a[n] = [a1[n], . . . , aK[n]]T and

z[n] = [z1[n], . . . , zK[n]]T where [·]T denotes transpose operation. Then (2.17) can be

equivalently written as

y[n] = a[n]s[n] + z[n]. (2.18)

Performance of communication systems employing the receive diversity technique de- pends on how the multiple signal copies are combined at the receiver. There are several ways of combining the received signals which vary in complexity and overall performance. According to the levels of channel state information (CSI) available at the receiver, there are three main combining methods, namely maximal-ratio combin- ing (MRC), equal-gain combining (EGC) and selection combining (SC) [R21, R25].

Maximal Ratio Combining

In this method, each individual received signal must be co-phased, weighted with its respective amplitude and then added up. The method is called optimum combining (regardless of the fading statistics) in the sense that it maximizes the received signal- to-noise ratio, γ, of the system under Gaussian noise. The maximal SNR is equal to the sum of all the instantaneous SNRs of the individual signals, i.e.,

γ =||a[n]||¯¯γ, (2.19)

where ¯¯γ = Es/N0 (note that the mean value of γ is ¯γ = ΩEs/N0) and ||a[n]|| =

PK

complexity since all the fading channel parameters need to be available at the receiver [R21].

Equal Gain Combining

As mentioned, MRC requires the knowledge of the time-varying channel coeffi- cients including amplitudes and phases of all branches. A suboptimal method, called equal-gain combining (EGC) with coherent detection, is an attractive alternative since it does not require the channel amplitude estimation. Hence, the implemen- tation complexity relative to the optimal MRC is reduced. In this method, all the received signals are co-phased and simply added together. The received SNR in this method can be calculated as

γ = 1 K K X k=1 q |ak[n]|2¯¯γ !2 (2.20)

Furthermore, the performance of EGC is only marginally inferior to that of the opti- mal MRC [R25].

Selection Combining

Selection combining is even a simpler diversity combining method. In this ap- proach, the receiver picks up the signal with the highest SNR (or equivalently, with the strongest incoming path assuming equal noise power in all branches) for detection. The output of the SC combining is computed as

γ = 

max

k |ak[n]|

2_{¯¯γ (2.21)}

Moreover, since the output of the SC detector involves only one of the branches, SC can be employed with differentially coherent and noncoherent modulation techniques [R21]. In practice, the signal branch with the highest sum of signal and noise power is often used as it is more difficult to measure the SNR.

Transmit Diversity

Transmit diversity has recently been studied extensively as an effective method to overcome the detrimental effects in wireless communications. Transmit diversity

can reduce the required signal processing effort of the receiver, which leads to a lower systems’ complexity, power consumption and cost [R5,R6,R26–R28]. Moreover, when incorporated with receive diversity, transmit diversity further improves the system performance. Different from receive diversity, it is more difficult to exploit spatial diversity with transmit diversity. This is because a careful design of the transmitted signals is required at the transmitter in order to distinguish the received signals at the receiver and exploit diversity since the transmitted signals are spread out to all transmit antennas; and the transmitter does not generally have the instantaneous CSI (unless the information is fed back from the receiver to the transmitter). To deal with these difficulties, a number of the transmit diversity schemes have been proposed in the literature [R29, R30]. An equivalent approach to realize transmit diversity is to view coding, modulation and multiple transmission as one signal processing module [R27]. Such an approach for multiple transmit antennas is called space- time coding. In particular, the transmitted signals are spread over both spatial and temporal dimensions, which introduces correlations over the transmitted signals.

In what follows, we discuss one of the particularly simple and yet elegant space- time codes to gain some insight into how a space-time coding system works. It is the very well-known Alamouti scheme, originally designed for 2 transmit antennas [R26]. The generalization to more than 2 antennas can be done under some restrictions [R6,R27]. The Alamouti scheme works over two consecutive symbol time slots under the assumption that channel coefficients are constant during that period, i.e., n and (n+1) time slots. During the nth time slot, two signals s1[n] and s2[n] are transmitted

simultaneously from antenna one and antenna two, respectively. In the (n+ 1)th time slot, signals _−s∗

2[n] and s∗1[n] are respectively sent from antenna one and antenna

two, where (·)∗ _{is complex conjugate operation. Denote the channel coefficients from}

antenna one and antenna two to the receiver by a1[n] and a2[n], respectively. The

input-output relation can be then written in matrix form as:

 y[n] y[n + 1]  =  a1[n] a2[n]     s1[n] −s∗2[n] s2[n] s∗1[n]   +  z[n] z[n + 1]  , (2.22)

or    y[n] y∗_{[n + 1]}    | {z } y[n] =    a1[n] a2[n] a∗ 2[n] −a∗1[n]    | {z } A[n]    s1[n] s2[n]    | {z } s[n] +    z[n] z∗_{[n + 1]}    | {z } z[n]