16.2.1 System Model

Assume slowly varying frequency-flat wireless channels with additive white Gaussian noise (AWGN). For the SISO channel, this means that the discrete-time equivalent lowpass model (Figure 16.2) at time $k\in \mathbb{Z}$ is

where $y(k)\in ℂ$ are the received samples, $x(k)\in ℂ$ are the transmitted symbols (e.g., belonging to M-ary quadrature amplitude modulation (M-QAM) or phase shift keying (PSK) constellations), $n(k)~\mathcal{C}\mathcal{N}(0,{\sigma }^2)$ are the noise samples, assumed independent, identically distributed (i.i.d.) circularly symmetric complex Gaussian (CSCG) random variables (r.v.s) with^* $\mathbb{E}\left\{|n(k){|}^2\right\}={\sigma }^2$. The channel gain $h\in ℂ$ is assumed constant over several symbols. Note that multiplication of the symbols x(k) by a complex gain h means that the constellation is scaled by |h| and rotated of an angle φ = arg h.

The SNR conditional to h is^†

where P_t is the average transmitted power, and the Shannon capacity is

achieved when the symbols x(k) are i.i.d. CSCG. A different formula would hold when the symbols x(k) are constrained to belong to a specific constellation.

For ease of notation in the following we will drop, when possible, the time index k.

16.2.2 Fading, Ergodic Capacity, Outage Capacity

Assume now a fading channel, where h varies in time. In the so-called block fading channel (BFC) model, the channel h remains constant over a block of several symbols, taking i.i.d. values across different blocks. Thus, the fading process is specified by the block length (characterizing the time-correlation of the fading process) and by the distribution of the fading level within a block (e.g., h is a CSCG r.v. for a Rayleigh fading channel). In the following, we indicate with h(m) the fading level for the mth block.^* The discrete-time process $\cdots ,h\left(\text{1}\right),h\left(\text{2}\right),\cdots$ is ergodic (time averages equal ensemble averages) if the block duration is finite, and nonergodic if a realization is composed of just one block, that is, one fading level, lasting forever (in practice, for the duration of the communication).

*Note that circular symmetry implies zero mean, so $\mathbb{E}\left\{n(k)\right\}=0$.

†This SNR can be shown to be the ratio between the received energy per symbol and N0, conditional to a channel gain h, where N₀ is the one-sided noise spectral density.

In Figure 16.3, we report a realization |h(m)|²P_t/σ² for m = 0, … , 50 of a Rayleigh BFC (dashed curves). We set P_t/σ² = 10, the channel gain has been normalized to give $\mathbb{E}\left\{|h(m){|}^2\right\}=1$, giving the average signal-to-noise ratio $\mathbb{E}\left\{\text{SNR}(m)\right\}=P_t\text{/}{\sigma }^2$. We observe that, due to Rayleigh fading, there are large fluctuations of the SNR values with m. The variations normalized to the average of the SNR are reported on the right plot. In the same figure, we show a realization of the SNR at the output of a SIMO system with N_R = 8 antennas and maximal ratio combiner (MRC) receive diversity described in Section 16.4.1. We observe two effects of MRC: first, the average SNR is increased by a factor 8; second, the normalized SNR fluctuations are less severe. The first effect is called “array gain” (improvement in the average SNR). The second is called “diversity gain” (reduced variance of the normalized SNR as reported on the right plot).

*The block index m should not to be confused with the time index k. The two are coincident only when the block length is one symbol.

The variation of the SNR in time produces a variation of the link quality (throughput, error rate, delay, etc.). For example, we show in Figure 16.4 an example of C(h(m)) from Equation 16.3 for a realization of the time-discrete process h(m). Hereafter, we refer to Equation 16.3 as mutual information, since the concept of capacity requires some care (below). In the same figure, we report the capacity of the unfaded AWGN channel, log₂(1 + SNR), and the average $\mathbb{E}\left\{C(h(m))\right\}$ where the average is taken with respect to the distribution of h(m). The difference between the unfaded capacity and $\mathbb{E}\left\{C(h(m))\right\}$ is relatively small (3.46 and 2.91 [bits/s/Hz], respectively). However, C(h(m)) shows large fluctuations around its average.

Recalling that the capacity is the maximum information rate that can be sent reliably across a noisy channel by using forward error correction (FEC) with sufficiently long codewords, we can distinguish two situations: the case where the codeword length is large with respect to a block (that we indicate as fast fading, or system with ideal time interleaving), and the case where the block duration is larger than the codeword length (slow fading, or system with no interleaving).

In this regard, $\mathbb{E}\left\{C(h)\right\}$ is called ergodic capacity with channel unknown at the transmitter, and represents the information rate that a system with constant transmit power can achieve by using FEC when each codeword spans many fading blocks, that is, with ideal time interleaving. In this case, the fluctuations of the channel are somewhat averaged out by time interleaving.

When the codeword length is of the order of the fading block (slow fading), and assuming the information rate R (information bits/s/Hz) at the channel input is constant, the mutual information (16.3) will be for some blocks smaller than R, and for some larger than R. From information theory we know that the codewords in the former case will not be correctly decoded,^* while for the latter the codewords will be successfully decoded, provided a sufficiently powerful FEC coding scheme is adopted. We can therefore define the channel outage probability as follows:

which represents a lower bound on the codeword error rate for coded systems with information rate R at the input of a slow-fading channel without interleaving. We can also define the outage capacity at codeword error rate P_target as the value R_out to use at the channel input to guarantee a target outage probability P_target

and the resulting effective rate is R_out (1 − P_out(R_out)) (correctly received [information bits/s/Hz] over many fading blocks). This is sometimes called outage rate.

Assume now that the channel state h(m) is available at the transmitter side, a situation indicated as CSIT. In this case, a possibility, for the slow-fading channel case, is to change the information rate R(m) (information bits/s/Hz) for each block m, adapting its value below C(h(m)), so that the received codeword can be successfully decoded. The information rate is adapted by changing the FEC code rate and modulation format. In practice, a predefined set of couples (codes, modulation format) can be used to discretize the information rate. The transmitter then chooses the couple based on C(h(m)). Over a large number N of blocks the transferred rate achievable is bounded by $1\text{}/\text{}N{\sum }_{m=1}^{N}C(h(m))$ which, for the law of large numbers, converges to $\mathbb{E}\left\{C(h)\right\}$. Thus, $\mathbb{E}\left\{C(h)\right\}$ is also the achievable rate for a rate time-adaptive system.

*Recall that, for rates above capacity, the error probability goes exponentially to 1.

Still assuming CSIT, if we can also adapt the transmit power on a block-by-block bases, we could spend, for a given time-average power, more energy on the blocks with better SNR, and less on those with lower SNR, according to the water-filling principle. This power and rate time-adaptive system has a slightly larger achievable rate than the rate time-adaptive system, but with some additional practical issues due to the power variation with time.

For both rate time-adaptive and power and rate time-adaptive techniques, the information rate (i.e., number of information bits per block) change from block to block. For delay-sensitive applications this can be a problem, and constant rate solutions are preferable. For example, the transmit power can be varied to maintain a constant SNR at the receiver, and consequently a constant rate (like power control in cellular systems, also called channel inversion).

These considerations will be extended to MIMO channels in Chapter 17.

16.3.1 MIMO Line-of-Sight Channel

Let us assume the transmission of a passband signal $\tilde{s}(t)=\Re \left\{s(t){\text{e}}^{j2\pi f_{0}t}\right\}$, where f₀ is the carrier frequency and s(t) is the equivalent lowpass of $\tilde{s}(t)$. We assume that the signaling interval is T, which means that the variation rate of s(t) is approximately 1/T. A delay τ on the passband signal introduces a delay and a phase variation on the equivalent lowpass, since $\tilde{s}(t-\tau )=\Re \left\{s(t-\tau ){\text{e}}^{-j2\pi f_0\tau }{\text{e}}^{j2\pi f_{0}t}\right\}$.

The MIMO channel with line-of-sight (LOS) propagation is depicted in Figure 16.5. We indicate as d_i,j and τ_i,j the path length and delay associated with the path from the jth transmit antenna to the ith receive antenna, respectively. We also indicate with $h_{i,j}\text{}=\left|h_{i,j}\right|{\text{e}}^{-j2\pi f_0{\tau }_{i,j}}$ the complex channel gain whose modulus is related to the path loss and antenna gains, and whose argument is related to the path delay. The equivalent lowpass of the signal at the output of the ith receive antenna is therefore

*Note that the rank of H is always limited by the minimum between N_T and N_R.

where n_i(t) is the thermal noise. Although the path lengths d_i,j are, strictly speaking, different, if the distance d between the transmitter and the receiver is much greater than the antenna elements spacing, the difference among them is negligible and we can write ${\tau }_{i,j}\simeq \tau ,\left|h_{i,j}\right|\simeq \left|h\right|$. Therefore, in the above hypotheses, we can write

where the $h_{i,j}\simeq \left|h\right|{\text{e}}^{-j2\pi f_0\tau }$ are all equal. Assuming matched filtering and sampling with proper timing, the discrete-time equivalent lowpass model at time $k\in \mathbb{Z}$ is therefore given by Equation 16.6 where the channel matrix $H\in {ℂ}^{N_T\times N_R}$ has entries h_i,j which are practically identical, and has therefore rank 1. Thus, for MIMO LOS channels with link length d large compared to the antenna elements spacing, the channel model is given in Equation 16.6 where the matrix H can be considered rank 1.

16.3.2 MIMO with a Point Scatterer

Consider now the situation in Figure 16.6, where the emitted radiation is scattered by a single point scatter. We assume that all delay differences are small compared to the signaling interval T. Thus, the channel model is still of the form (16.6) but with a channel matrix $H=h_{(S\to RX)}{h}_{(TX\to S)}^T$, where h_(TX→S) is the N_T × 1 propagation vector from the transmitter to the scatterer, and h_(S→RX) is the N_R × 1 propagation vector from the scatterer to the receiver. These two vectors can be expressed precisely from the geometry depicted in Figure 16.6. From basic algebra, we know that, since H is the product of two vectors, it is rank 1. Thus, with one point scatterer, the channel matrix H in Equation 16.6 is rank 1.

16.3.3 MIMO with Multiple Scatterers

Assume now several point scatterers with delay differences which are small compared to the signaling interval T. Then, by the previous argument, the channel model is still of the form (16.6) with a channel matrix

where $h_{\left(TX\to S_k\right)}$, $h_{\left(S_k\to RX\right)}$ are the N_T × 1 and N_R × 1 propagation vectors for kth scatterer. From basic algebra, we thus obtain that the rank of H is at most equal to the number of scatterers (scatterers with the same direction of departure or of arrival will not increase the channel matrix rank). For a very large number of scatterers, the sum in Equation 16.9 can be thought of as an integral.

16.3.4 Statistical MIMO Channel Models

Consider now the ensemble of all possible channel matrices H, when the relative positions of the scatterers are subject to small random variations, due, for example, to movement of the transmitter, of the receiver, or of the environment. In this case the evolution of H with k is described by a discrete-time random process, where the elements of the channel matrix are randomly varying. The rate of variation of H depends on the rate of variation of the environment, that is, on the mobility of transmitter, receiver, and scatterers. The channel H at a given time k can be considered a random matrix, that is, a matrix with elements that are random variables, and a realization of it can be interpreted as the channel for a specific snapshot of the varying environment [10].

Let us consider a stationary channel, so that the first-order description of H(k) does not depend on k. If we assume that the randomly placed scatterers introduce delay differences which are small compared to the signaling interval T, the system model is still given by Equation 16.6, where H has elements h_i,j described as random variables. When the number of independent scatterers is large, from the central limit theorem, the elements h_i,j are complex Gaussian r.v.s, and their joint distribution can be given by specifying all mean values, $\mathbb{E}\left\{H\right\}$, and all pairs correlation, that is, the matrix $\mathbb{E}\left\{\text{vec}(H)\text{vec}{(H)}^H\right\}$, where vec(H) is the (N_T · N_R) × 1 vector obtained by stacking the columns of H on top of one another.

Often, the correlation at the transmitter side (i.e., among the elements of a row of H) is the same for all rows, and the correlation at the receiver side (i.e., among the elements of a column of H) is the same for all columns. In this case, the correlation matrix can be factorized as $\mathbb{E}\left\{\text{vec}(H)\text{vec}{(H)}^H\right\}=R_T\otimes R_R,$ where ⊗ denotes Kronecker product, and R_T and R_R describe the correlation at the transmitter and receiver side, respectively. This Kronecker model implies that the correlation at the receiver (transmitter) does not affect the spatial properties of the signals at the transmitter (receiver).

The spatial correlation is in general related to the angular distribution of the rays and the antenna elements spacing. Roughly speaking, many scatterers around the receiver (transmitter) give low correlation at the receiver (transmitter) side, while if all paths at one side arrive from a small angle, the correlation on that side is large, implying an overall rank-deficient channel.

For rich scattering environments with no dominant paths, one common model consists in assuming the elements h_i,j as CSCG, implying $\mathbb{E}\left\{H\right\}=0$ (Rayleigh MIMO channels). Generally, if the spread of the angles of departure and of arrival are large,^* the correlation is also small and the h_i,j can be considered i.i.d. This is the Rayleigh uncorrelated MIMO channel model.

16.4.1 Receive Diversity: SIMO with CSIR (MRC)

Consider the SIMO scheme in Figure 16.7. The N_R-dimensional signal at the output of the receiving antennas at time k can be written as

where $h\in {ℂ}^{N_R}$ is a vector containing the complex gains between the transmit antenna and the N_R receive antenna, $x(k)\in ℂ$ is the transmitted symbol at time k, $n(k)\in {ℂ}^{N_R}$ is the noise vector at time k with i.i.d. CSCG entries and covariance matrix $R_n=\mathbb{E}\left\{n(k)n^H(k)\right\}={\sigma }^{2}I$.

Here, conditionally to hx(k), the vector y(k) is Gaussian, and the best linear estimate of the transmitted symbol is^†

with weights

Note that Equation 16.12 requires that the channel gains (amplitude and phase) are known at the receiver. So, for each antenna element, we need a channel estimator (e.g., based on pilot symbols periodically embedded in the data symbols). Linear combining of the received signals with weights of the type w = Kh, as in Equation 16.12, with K an arbitrary constant, is called Maximal Ratio Combiner (MRC).^‡ The SNR at the combiner output is

*This condition in some particular cases is not sufficient. An example is the so-called keyhole channel, where, despite scattering, all paths are forced to pass through an intermediate point like in Figure 16.6; thus, as in Section 16.3.2, the overall channel matrix is rank 1 since it can be expressed as the product of two vectors.

†This estimate produces a sufficient statistic for the detection of the transmitted symbol and, as we will see later, is capacity-optimal.

‡It is easy to show that weights in the form w = Kh give the maximum SNR after the combiner.

where P_t is the average transmitted power. It is easy to verify that, when the h_i are i.i.d., the array gain is N_R and the diversity order is also N_R.

The implementation of MRC is sketched in Figure 16.8.

16.4.2 MISO and Diversity with CSIT (Beamforming)

Consider the MISO scheme in Figure 16.10, and assume that the transmitter knows perfectly the channel. The objective is to improve the performance by sending on the transmit antennas a linearly precoded version of the symbol x(k).

The weights α₁, α₂ must be designed to focus the transmitted energy toward the receiver, to produce the maximum SNR at the receive antenna. The power at the ith transmit antenna is $\mathbb{E}\left\{|x(k){|}^2\right\}|{\alpha }_i{|}^2$, and we impose the normalization ${\sum }_i|{\alpha }_i{|}^2=|\left|\right|{|}^2=1$ to have an average total transmitted power $P_t=\mathbb{E}\left\{|x(k){|}^2\right\}$ as for the single transmit antenna case.

The received signal is

It is easy to see that the best SNR for y(k) is obtained by choosing ${\alpha }_i=K{h}_i^{*}$, where K is a normalization constant with $\left|K{|}^2{\sum }_i\right|h_i{|}^2=\left|K{|}^2\right|\left|h\right|{|}^2=1$. Taking K > 0 real (the phase of K can be included in the carrier phase), we have

The choice ${\alpha }_i=K{h}_i^{*}$ makes the signals at the receive antenna to add coherently, and the resulting SNR at the receiver is

which is exactly the same as for the MRC output (16.13).

This scheme can be called beamforming or eigenbeamforming and gives, for MISO with CSIT, exactly the same performance as for a SIMO MRC with CSIR (for the same total power P_t and channel gains h_ℓ). In particular, the array gain is N_T and the diversity order is N_T.

Different from MRC, here the transmitter must know the channel (amplitudes and phases for all h_l). The ways to obtain this information are essentially two:

• If the system adopts a time division duplex (TDD) technique, the channel estimated in reception (e.g., by using pilot symbols) can be used, for reciprocity, as an estimate of the channel in transmission (assuming the channel does not vary too fast).

• In closed-loop systems, the receiver estimates the channel and sends the estimates back to the transmitter. This requires a feedback channel, with problems related to delays, quantization, and transmission errors.

16.4.3 MIMO Beamforming or Eigenbeamforming (Single Beam)

The previous approach can be easily extended to the case when both the TX and the RX are equipped with multiple antenna (Figure 16.11). We call this MIMO beamforming or eigenbeamforming (single beam).

The received vector is

where $\tilde{h}=H$. Since the last expression looks like Equation 16.10 for SIMO channels, we have that the optimum weights to use in reception are immediately given by the MRC principle as

where K is an arbitrary constant and the corresponding SNR is

For the weights in transmission, we have to find the vector α, with $\Vert a\Vert =1$, which maximizes Equation 16.22, that is, maximizing $|\left|\tilde{h}\right|{|}^2={\tilde{h}}^H\tilde{h}=a^H{H}^{H}Ha.$. Since H^HH is Hermitian, we can use the spectral decomposition H^HH = VΛV^H, where the columns of the unitary matrix V are the eigenvectors of H^HH, and Λ is diagonal with elements on the diagonal that are the eigenvalues of H^HH. From linear algebra we recall that the vector α maximizing α^HVΛV^Hα is the eigenvector of H^HH corresponding to the largest eigenvalue λ_max of H^HH. In this case, it results as α^HVΛV^Hα = λ_max.

Thus, in summary

• The optimum precoding vector α_opt is the eigenvector of H^HH corresponding to the largest eigenvalue λ_max of H^HH

• The optimum combining vector is w = KHα_opt

• The resulting SNR at the output of the MIMO beamforming is

16.4.4 MISO and Diversity without CSIT, with CSIR (Alamouti's Scheme, Space–Time Codes)

We have seen that MISO beamforming requires CSIT, but is it possible to have transmit diversity without CSIT? Even looking at the simple 2 × 1 MISO it is quite clear that, if the TX does not know the channel, it is not possible by using a beamforming-like approach to find weights able to build distinct channels (and thus to provide diversity). However, it has been shown that by using, besides the spatial, the time dimension, it is actually possible to have transmit diversity even without CSIT. We discuss below the Alamouti's scheme for 2 × 1 MISO.

While in previous schemes we assumed to transmit a symbol x(k) (in a given time symbol T) and our aim was to recover that symbol, the key idea in space–time codes is to use time to build separable channels. So, assume we look at two consecutive symbols, x(k), x(k + 1), to be transmitted in two symbol periods. In a first symbol period T, we transmit x(k) on the first antenna and x(k + 1) on the second. In the second period, we transmit x*(k + 1) on the first antenna and −x*(k) on the second, as reported in Figure 16.12. The symbols are “precoded” by a factor $1\text{/}\sqrt{2}$ to maintain $P_t=\mathbb{E}\left\{|x{|}^2\right\}$.

At the receiver, the two samples received in the two consecutive time instants k, k + 1 are

By conjugating the second equation and defining $\tilde{y}(k)={(y(k),y^{*}(k+1))}^T,\tilde{n}(k)={(n(k),n^{*}(k+1))}^T,\text{x}\left(k\right)={\left(x\left(k\right),x\left(k+1\right)\right)}^T$, we can write

or, in short, $\tilde{y}(k)=\frac{1}{\sqrt{2}}Ax(k)+\tilde{n}(k)$, where

Now, by using time, with this construction we have in Equation 16.25 a virtual 2 × 2 MIMO. In principle, assuming the receiver knows h and therefore A (this means CSIR), the symbols x(k), x(k + 1) can be detected by using, for example, maximum likelihood (ML). However, here the situation is even simpler, since we observe that

that is, the matrix A is proportional to a unitary matrix. Therefore, if we compute $ŷ(k)=1\text{}/\text{}\left|\right|h\left|\right|A^H\tilde{y}(k),$ we have

where it is easy to verify that the Gaussian vector $̂̂̂n(k)={(n(k),n(k+1))}^T=1\text{}/\text{}\left|\right|h\left|\right|A^H\tilde{n}(k)$ has $̂\mathbb{E}\left\{n(k)\right\}=0$ and $̂̂\mathbb{E}\left\{n(k)n^H(k)\right\}=1\text{}/\text{}\left|\right|h|{|}^2{\sigma }^2{A}^{H}A={\sigma }^{2}I$, and is therefore statistically equivalent to the original noise n(k) = (n(k),n(k + 1))^T. Therefore, we have the equivalent space–time channel

In other words, with this approach, the 2 × 1 MISO without CSIT can be interpreted as the SISO in Figure 16.13. In particular, we do not need to rethink to specific modulation and coding, since we can apply here all modulation and coding/decoding techniques for the SISO channel.

The virtual SISO channel obtained by the Alamouti's scheme has output SNR

Comparing this expression with Equation 16.19, we see that this 2 × 1 MISO scheme with no CSIT has the same performance of a 2 × 1 MISO with perfect CSIT and beamforming, except for a 3 dB loss (factor 2) in SNR. This loss is due to the lack of CSIT which does not allow precoding, like in beamforming, to have the coherent sum of signals at the receive antenna (here the weights in transmission are both equal to $1\text{/}\sqrt{2}$, meaning that the energy is transmitted uniformly in all directions).

In fact, the array gain is 1 (no array gain), while the diversity gain is N_R = 2.

It is also easy to show that, for a 2 × N_R MIMO Alamouti's scheme with channel matrix $H\in {ℂ}^{2\times N_R}$, we can replicate the scheme above for each receiving antenna (rows of H) and add the outputs. The SNR at the output of the combiner is the sum of the corresponding SNRs, that is,

From this expression, we see that the Alamouti's scheme uses the full 2 · N_R diversity available.

This scheme, specifically designed for N_T = 2, has the same spectral efficiency of a SISO, in the sense that two symbols are transmitted in two symbol periods. In this regard, the encoder used (see $\mathcal{C}$ in Figure 16.13) is rate 1.

For N_T > 2 it has been proved that similar orthogonal codes can be designed, but with rates strictly smaller than 1.

More in general, schemes like that in Figure 16.13, with suitable decoders, lead to the so-called space–time codes, where we can have block encoders (space–time block codes (STBC)), or trellis encoders (space–time trellis codes (STTC)). The Alamouti's scheme can be considered an STBC with N_T = 2 and orthogonal codewords.

16.4.5 Performance of Alamouti's Scheme

The Alamouti's scheme can be used with coding and modulation designed for SISO channels, so its performances are those of a SISO system, with the SNR given by Equation 16.30. For example, in Figure 16.14, we report the BEP for uncoded Alamouti's scheme with M-QAM over a (2 × 2) MIMO Rayleigh uncorrelated fading channel with $\mathbb{E}\left\{|h_{i,j}{|}^2\right\}=1$, as a function of E_b/N₀ where E_b is the energy transmitted per information bit (see also Chapter 17).