If feedback is added to the system, the transmitter may be able to have knowledge of the channel between it and the receiver. Because the channel changes quickly in a highly mobile scenario, closed-loop transmission schemes tend to be feasible primarily in fixed or low-mobility scenar- ios. As we shall see, however, there is a substantial gain in many cases from possessing channel state information (CSI) at the transmitter, particularly in the spatial multiplexing setup discussed
Na = 6
2×4 1 5×
1 5×
γ1 5MRC× = 5γ
γ2 4 =1 γ γ 28 = 4
× STBC
2×4 5 1×
2×4 ×
later in the chapter. This again has motivated intensive research on techniques for achieving low- rate prompt feedback, often specifically for the multiantenna channel [42].
The basic configuration for closed-loop transmit diversity is shown in Figure 5.8; in gen- eral, the receiver could also have multiple antennas, but we neglect that here for simplicity. An encoding algorithm is responsible for using the CSI to effectively use its available channels.
We will assume throughout this section that the transmitter has fully accurate CSI available to it, owing to the feedback channel. We now review two important types of closed-loop transmit diversity, focusing on how they affect the encoder design and on their achieved performance.
5.3.3.1 Transmit Selection Diversity
Transmit selection diversity (TSD) is the simplest form of transmit diversity, and also one of the most effective. In transmit selection diversity first suggested by Winters [65], only a subset of the available antennas are used at a given time. The selected subset typically corresponds to the best channels between the transmitter and the receiver. TSD has the advan- tages of (1) significantly reduced hardware cost and complexity, (2) reduced spatial interference, since fewer transmit signals are sent, and (3) somewhat surprisingly, diversity order, even though only of the antennas are used. Despite its optimal diversity order, TSD is not optimal in terms of diversity gain.
In the simplest case, a single transmit antenna is selected, where the chosen antenna results in the highest gain between the transmitter and the receive antenna. Mathematically, this is sta- tistically identical to choosing the highest-gain receive antenna in a receive-diversity system, since they both result in an optimum antenna choice :
Figure 5.7 Comparison of the Alamouti STBC with MRC for coherent BPSK in a Rayleigh fading channel.
−5 0 5 10 15 20 25
10−6 10−5 10−4 10−3 10−2 10−1 100
AverageSNR (dB)
Average Bit Error Probability N
a=2 (No Diversity) Na= 3
Na= 4 Na= 5
Na= 6
MRC (N t= 1) - - - STBC (N
t= 2)
Na : Total Number of Transmitand Receive Antennas
_____
Nt
N*<Nt Nt
N Nt r
N* Nt
i*
(5.34) Hence, TSD does not incur the power penalty relative to receive selection diversity that we observed in the case of STBCs versus MRC, while achieving the same diversity order. The aver- age SNR with single-transmit antenna selection in a system with i.i.d. Rayleigh fading is thus
(5.35) which is identical to Equation (5.13) for receiver selection combining. This is, however, a lower average SNR than can be achieved with beamforming techniques that use all the transmit anten- nas. In other words, transmit selection diversity captures the full diversity order—and so is robust against fading—but sacrifices some overall SNR performance relative to techniques that use or capture all the available energy at the transmitter and the receiver.
The feedback required for antenna transmit selection diversity is also quite low, since all that is needed is the index of the required antenna, not the full CSI. In the case of single-transmit antenna selection, only bits of feedback are needed for each channel realization. For example, if there were transmit antennas and the channel coherence time was msec—corresponding to a Doppler of about 100Hz—only about 1kbps of channel feed- back would be needed, assuming that the feedback rate was five times faster than the rate of channel decorrelation.
In the case of active transmit antennas, choosing the best out of the available elements requires a potentially large search over
(5.36) Figure 5.8 Closed-loop transmit diversity
Receiver Transmit
Diversity Encoder
h1
h2
hN
Feedback Channel, {h1,h2, ... hN}
t
t
i h
i Nt i
*
(1,
=arg max )| | .2
∈
Nt ×1
γtsd γ
i Nt
= 1i ,
∑=1
log2Nt Nt = 4 Tc = 10
N* N* Nt
( )NN*t
different possibilities, although for many practical configurations, the search is simple. For example, choosing the best two antennas out of four requires only six possible combinations to be checked. Even for very large antenna configurations, near-optimal results can be attained with much simpler searches. The required feedback for transmit antenna selection is about bits per channel coherence time. Because of its excellent performance versus com- plexity trade-offs, transmit selection diversity appears to be attractive as a technique for achiev- ing spatial diversity, and has also been extended to other transmit diversity schemes such as space/time block codes [12, 31], spatial multiplexing [33], and multiuser MIMO systems [9]. An overview of transmit antenna selection can be found in [45].
In the context of WiMAX, a crucial drawback of transmit antenna selection is that its gain is often very limited in a frequency-selective fading channel. If the channel bandwidth is much wider than the channel coherence bandwidth, considerable frequency diversity exists, and the total received power in the entire bandwidth will be approximately equal regardless of which antenna is selected. If each OFDM subcarrier were able to independently choose the desired transmit antenna that maximized its subcarrier gain, TSD would be highly effective, but sending a different subset of subcarriers on each transmit antenna defeats the main purpose of transmit antenna selection: turning off (or not requiring) the RF chains for the Nt – N* antennas that were not selected. Additionally, in this case, the required feedback would increase in proportion to L (the number of subcarriers). Hence, despite its theoretical promise, transmit selection diversity is likely to be useful only in deployments with small bandwidths and small delay spreads (low range), which is very limiting.
5.3.3.2 Linear Diversity Precoding
Linear precoding is a simple technique for improving the data rate, or the link reliability, by exploiting the CSI at the transmitter. In this section, we consider diversity precoding, a special case of linear precoding whereby the data rate is unchanged, and the linear precoder at the trans- mitter and a linear postcoder at the receiver are applied only to improve the link reliability. This will allow comparison with STBCs, and the advantage of transmit CSI will become apparent.
With linear precoding, the received data vector can be written as
(5.37) where the sizes of the transmitted (x) and the received (y) symbol vectors are , the post- coder matrix G is , the channel matrix H is , the precoder matrix F is , and the noise vector n is . For the case of diversity precoding (comparable to a rate 1 STBC), , and the SNR maximizing precoder F and postcoder G are the right- and left- singular vectors of H corresponding to its largest singular value, . In this case, the equivalent channel model after precoding and postcoding for a given data symbol is
(5.38) N*log2Nt
y=G HFx( +n),
M×1
M×Nr Nr×Nt Nt×M
Nr×1 M= 1
σmax M= 1 x y=σmax⋅ +x n.
Sid e b ar 5. 1 A B r i ef Pr i me r on Ma t r i x T h eo r y
As this chapter indicates, linear algebra and matrix analysis are an inseparable part of MIMO theory. Matrix theory is also useful in understanding OFDM. In this book, we have tried to keep all the matrix notation as standard as possible, so that any appropriate reference will be capable of clarifying any of the presented equations.
In this sidebar, we simply define some of the more important notation for clarity.
First, in this chapter, two types of transpose operations are used. The first is the conven- tional transposeAT, which is defined as
that is, only the rows and columns are reversed. The second type of transpose is the con- jugate transpose, which is defined as
.
That is, in addition to exchanging rows with columns, each term in the matrix is replaced with its complex conjugate. If all the terms in A are real, AT = A*. Sometimes, the conjugate transpose is called the Hermitian transpose and denoted as AH. They are equivalent.
Another recurring theme is matrix decomposition—specifically, the eigendecompo- sition and the singular-value decomposition, which are related to each other. If a matrix is square and diagonalizable (M× M), it has the eigendecomposition
A = TΛT–1,
whereT contains the (right) eigenvectors of A, and Λ = diag[λ1λ2 ... λM] is a diagonal matrix containing the eigenvalues of A.T is invertible as long as A is symmetric or has full rank (M nonzero eigenvalues).
When the eigendecomposition does not exist, either because A is not square or for the preceding reasons, a generalization of matrix diagonalization is the singular-value decomposition, which is defined as
A = UΣV*,
whereU is M× r,V is N× r, and Σ is r× r, and the rank of A—the number of nonzero singular values—is r. Although U and V are no longer inverses of each other as in eigen- decomposition, they are both unitary—U*U = V*V = UU* = VV* = I—which means that they have orthonormal columns and rows. The singular values of A can be related to eigenvalues of A*A by
.
BecauseT–1 is not unitary, it is not possible to find a more exact relation between the sin- gular values and eigenvalues of a matrix, but these values generally are of the same order, since the eigenvalues of A*A are on the order of the square of those of A.
AijT = Aji
Aij* = (Aji)*
σi( )A = λi(A*A)
Therefore, the received SNR is
(5.39) whereσ2 is the noise variance. Since the value or expected value of σmax is not deterministic, the SNR can be bounded only as [47],
(5.40) where denotes the Frobenius norm and is defined as
(5.41) On the other hand, by generalizing the SNR expression for STBCs—Equation (5.24)—
the SNR for the case of STBC is given as
(5.42) By comparing Equation (5.40) and Equation (5.42), we see that linear precoding achieves a higher SNR than the open-loop STBCs, by up to a factor of . When , the full SNR gain of dB is achieved; that is, the upper bound on SNR in Equation (5.40) becomes an equality.
To use linear precoding, feeding back of CSI from the receiver to the transmitter is typically required. To keep the CSI feedback rate small, a codebook-based precoding method that requires only 3–6 bits of CSI feedback for each channel realization has been defined for WiMAX. More discussion on codebook-based precoding can be found in Section 5.8, with WiMAX implemen- tations discussed in Chapter 8.
5.4 Beamforming
In contrast to the transmit diversity techniques of the previous section, the available antenna ele- ments can instead be used to adjust the strength of the transmitted and received signals, based on their direction, which can be either the physical direction or the direction in a mathematical sense.
This focusing of energy is achieved by choosing appropriate weights for each antenna element with a certain criterion. In this section, we look at the two principal classes of beamforming: direc- tion of arrival (DOA)–based beamforming (physically directed) and eigenbeamforming (mathe- matically directed). It should be stressed that beamforming is an often misunderstood term, since these two classes of “beamforming” are radically different.
γ=εσ σ ,
2 x 2
max
H F H
2 x
F 2
Nt
⋅ε ≤ ≤ ⋅εx
σ γ
σ
2 2,
⋅ F
H F= .
=1 =1 2 i Nt
j Nr
hij
∑∑
2 2×
γ σ
ε
STBC
x
Nt
= H F22 .
Nt Nr = 1 10log10Nt