Coding and Signal Processing for Multiple-Antenna Transmission Systems: A Review Ezio Biglieri∗ , Giorgio Taricco∗ , Antonia Tulino‡ ∗
Politecnico di Torino Corso Duca degli Abruzzi 24 I-10129 Torino (Italy) Email: @polito.it ‡
Universit`a del Sannio Palazzo dell’Aquila Bosco Lucarelli Corso Garibaldi 107 I-82100 Benevento (Italy) Email:
[email protected] Abstract We describe recent results on multiple-antenna transmission systems. These are known to offer a potential performance, in terms of bit/s/Hz, that cannot be obtained otherwise with present-day technology. We summarize the theoretical performance limits of these systems under the assumptions that the channel gains are known to the receiver, to the transmitter and to the receiver, or to neither. Coding and signal-processing schemes that allow these limits to be approached are finally addressed.
Introduction Recent work (see [2] and references therein) has explored the ultimate performance limits of systems with multiple transmit and receive antennas in a fading environment. These suggest the potential for considerable improvement in spectral efficiencies, to a level that cannot be achieved by any other technique with present-day technology. Several informationtheoretic analyses have been published: for example, [32] contains an analysis of several practical transmission schemes with multiple antennas, while [13, 14, 38] determine the information capacity of multiple-antenna channels. The communication link comprises t transmit and r receive antennas operating on a flat-fading channel. The channel gains are described by a r × t random matrix H of complex coefficients, and the additive channel noise is statistically independent across the r receivers. The standard fading channel model for a multiple-antenna system is one in which a block of several adjacent symbols (their number is denoted by I in the following) is affected by the same fading value, and different blocks are affected by independent fading values. For example, this model is applicable to an indoor wireless data network or a personal communication system with mobile terminals moving at walking speed, so that the channel gain, albeit random, varies so slowly with time that it can be assumed as constant along a block (see also [12–14, 34, 37, 38]). More generally, fading blocks can be thought of as separated in time (e.g., in a time-division system [33]), as separated in frequency (e.g., in a multicarrier system), or as separated both in time and in frequency (e.g., with slow time-frequency hopping [8,23,24]). This model, denoted the block-fading (BF) channel, allows the incorporation of delay constraints, which imply that, even though very long code words are transmitted, perfect (i.e., infinite-depth) interleaving cannot be achieved. With this fading model, the channel may or may not be ergodic (ergodicity is lost when there is a delay constraint). Without ergodicity, the average mutual information over the ensemble of channel realizations cannot characterize the achievable transmission rates: since the instantaneous mutual information of the channel turns out to be a random variable, the quantity that characterizes the quality of a channel when coding is used is not channel capacity, but rather the information outage probability. i.e., the probability that this mutual information is lower than the rate of the code used for transmission [33]. This outage probability is closely related to the code word error probability, as averaged over the random coding ensemble and over all channel realizations; hence, it provides useful insight on the performance of a delay-limited coded system [8, 23, 24, 26, 27]. An additional important definition related to outage probability is that of delay-limited capacity, sometimes also referred to as zero-outage capacity. This is the maximum rate for which the minimum outage probability is zero under a given power constraint [1, 10, 11].
Influence of channel-state information A crucial factor in determining the performance of a multiple-antenna system is the availability of the channel-state information (CSI), that is, the knowledge of the fading gains in each one of the transmission paths, that is, of the values taken on by the entries of the matrix H. For example, in a system with t transmit and r receive antennas and an ergodic Rayleigh fading channel modeled by an t × r matrix with random i.i.d. complex Gaussian entries the average channel capacity with perfect CSI at the receiver is about m , min(t, r) times larger than that of a single-antenna system for the same transmitted power and bandwidth. The capacity increases by about m bit/s/Hz for every 3-dB increase in signal-to-noise ratio (SNR). Due to the assumption of perfect CSI available at the receiver, this result can be viewed as a fundamental limit for coherent multiple-antenna systems [39]. In a fixed wireless environment, the fading gains can be expected to vary slowly, so their estimate can be obtained by the receiver with a reasonable accuracy, even in a system with a large number of antennas. One way of obtaining this estimate is by periodically sending pilot signals on the same channel used for data signals: if the channel is assumed to remain constant for I symbol periods, then we may write I = Ir + Id , where Ir is the number of pilot symbols, while data transmission occupies Id symbols. Since the transmission of pilot symbols lowers the information rate, there is a tradeoff between system performance and transmission rate. No channel state information. The fundamental limits of non-coherent communication, i.e., one taking place in an environment where estimates of the fading coefficients are not available, were derived by Marzetta and Hochwald [20,30]. In their channel model the fading gains are i.i.d. Rayleigh and remain constant for I symbol periods before changing to new independent realizations. Under these assumptions, they proved that further increasing the number of transmit antennas beyond I cannot increase capacity. Zheng and Tse [39] derive an explicit formula for the high-SNR average channel capacity for t = r and I ≥ 2r, and characterize the rate at which capacity increases with SNR for I < 2r. By defining K , min(r, t, bI/2c), [39] shows that the capacity gains K(1 − K/I) bits per second per Hz for every 3-dB SNR increase. In [39] it is also shown that the capacity is maximized by using no more than K transmit antennas: in particular, having t > r does not improve capacity. The same high-SNR capacity behavior can be achieved by sending K training symbols to estimate the fading coefficients, and using a mismatched receiver which treats the estimated channel as if it were the true channel. Perfect CSI at the receiver. The most commonly studied situation is that of perfect CSI available at the receiver. Among the results of [38], it is shown that, in a system with t transmit and r receive antennas and a slow fading channel modeled by an t × r matrix with random i.i.d. complex Gaussian entries (the “independent Rayleigh fading” assumption), the average channel capacity with perfect channel-state information (CSI) at the receiver is about m , min(t, r) times larger than that of a single-antenna system for the same transmitted power and bandwidth. The capacity increases by about m bit/s/Hz for every 3-dB increase in signal-to-noise ratio (SNR). Moreover, it is shown in [38] that a limited form of reciprocity holds between transmit- and receive-antenna diversity: specifically, if C(r, t, P) denotes the capacity of a channel with r receivers, t transmitters and total transmit power P, then for a deterministic (i.e., AWGN) channel we have C(a, b, P) = C(b, a, P), while for the random channel considered here C(a, b, Pb) = C(b, a, Pa). Under a non-ergodic channel assumption, refs. [13, 14, 16, 38] investigate the outage probability of a single-block block-fading channel with CSI at the receiver. Imperfect CSI at the receiver. In real world the receiver has an imperfect knowledge of the CSI. In [28] answers are provided to two related questions, referred to a transmission architecture to be described later on and called BLAST: 1) How long should the training interval be for satisfactory operation? 2) What effects do estimation errors have on performance? It is shown that we must have It ≥ t, i.e., the duration of the training interval must be at least as great as the number of transmit antennas. Moreover, the optimum training signals are orthogonal with respect to time among the transmit antennas, and each transmit antenna is fed equal energy. As for the effects of channel estimation errors, the training interval required to control the probability of outages induced by these errors is approximately proportional to the number of receive antennas. Thus, if the total interval I for training and for transmitting data is limited, the situation is the following. If we assume t = r for simplicity, the capacity in units of bits per symbol is approximately proportional to t: thus Ir = αt, for some constant α, and this leaves
I − Ir = I − αt symbols for sending the message. The capacity is C = βt for some constant β, so the total number of bits that can be sent is CId = βt(I − αt). The optimum number of transmit antennas from the standpoint of maximizing the number of message bits is t = I/2α, implying that, whatever the values of α and β, the training interval is Ir = I/2: that is, half of the available interval should be used for training [28]. CSI at the transmitter and at the receiver. It is also possible to envisage a situation in which channel state information is known to the receiver and to the transmitter: the latter can take the appropriate measures to counteract the effect of channel attenuations by suitably modulating its power. To assure causality, the assumption of CSI available at the transmitter is valid if it is applied to a multicarrier transmission scheme in which the available frequency band (over which the fading is selective) is split into a number of subbands, as with OFDM. The subbands are so narrow that fading is frequency-flat in each of them, and they are transmitted simultaneously, via orthogonal subcarriers. From a practical point of view, the transmitter can obtain the CSI either by a dedicated feedback channel (some existing systems already implement a fast power-control feedback channel) or by time-division duplex, where the uplink and the downlink timeshare the same subchannels and the fading gains can be estimated from the incoming signal. Ref. [1] derives the performance limits of a channel with additive white Gaussian noise, delay and transmit-power constraints, and perfect channel-state information available at both transmitter and receiver. Because of a delay constraint, the transmission of a code word is assumed to span a finite (and typically small) number M of independent channel realizations; therefore, the channel is nonergodic, and the relevant performance limits are the information outage probability and the delay-limited capacity. The coding scheme that minimizes the information outage probability is also derived in [1]. This scheme can be interpreted as the concatenation of an optimal code for the AWGN channel without fading to an optimal linear beamformer, whose coefficients change whenever the fading changes. For this optimal scheme minimum-outage probability and delay-limited capacity were evaluated. Among other results, [1] proves that, for the fairly general class of regular fading channels, the asymptotic delay-limited capacity slope, expressed in bit/s/Hz per dB of transmit SNR, is proportional to m , min(t, r) and independent of the number of fading blocks M . Since M is a measure of the time diversity (induced by interleaving) or of the frequency diversity of the system, this result shows that, if channel-state information is available also to the transmitter, very high rates with asymptotically small error probabilities are achievable without need of deep interleaving or high frequency diversity. Moreover, for a large number of antennas the delay-limited capacity approaches the ergodic capacity. Finally, the availability of CSI at the transmitter makes transmit-antenna diversity equivalent, in terms of capacity improvement, to receive-antenna diversity.
Coding for multiple-antenna systems The design of codes for multiple-antenna systems (“space-time codes” [37]) is attracting considerable attention. Assume that a space-time code is used with block length N . The transmitted signal is represented by the t × N matrix X , (x[1], . . . , x[N ]). The code, which we denote X, has |X| words. The row index of X indicates space, while the column index indicates time: to wit, the ith component of the t-vector x[n], denoted xi [n], is a complex number describing the signal transmitted by the ith antenna at discrete time n (for i = 1, . . . , t, n = 1, . . . , N ). The received signal is the r × N matrix Y = HX + Z (1) where Z is matrix of zero-mean complex Gaussian random variables (RV) with zero mean and independent real and imaginary parts with the same variance N0 /2 (i.e., circularly-distributed). Thus, the noise affecting the received signal is spatially and temporally independent, with E[ZZ† ] = N N0 Ir , where Ir denotes the r × r identity matrix and (·)† denotes Hermitian transposition. The channel is described by the r × t matrix H whose complex entries hij are circularly distributed with variance of their real and imaginary parts equal to 1/2r. Equivalently, each entry of H has uniformly-distributed phase and Rayleigh-distributed magnitude, with expected magnitude square equal to 1/r [38]. H is independent of both X and Z, it remains constant during the transmission of an entire code word, and its realization (the “channel state information,” or CSI) is known at the receiver. Notice that the normalizing factor 1/r introduced in the variance of the elements of H in (1) ensures that the total power received by r antennas from each transmit antenna remains constant as r varies (and, in particular, it does not diverge as r → ∞). Under the assumption of CSI perfectly known by the receiver, and of additive white Gaussian noise, maximumlikelihood detection and decoding corresponds to choosing the code word X which minimizes the squared “Frobenius”
norm kY − HXk2 , where we define, for an m × n matrix A with elements aij : 2
kAk ,
m X n X
|aij |2 = Tr(AA† ) = Tr(A† A)
(2)
i=1 j=1
Explicitly, ML detection and decoding corresponds to the minimization of the quantity 2 N X r t X X 2 kY − HXk = y − h x in ij jn n=1 i=1
(3)
j=1
For practical computation of error probabilities, it is customary to resort to the union bound to error probability 1 X X b P (X → X) P (e) ≤ |X|
(4)
X∈X X∈X\{X} b
b is given by where the pairwise error probability P (X → X) !# " h i b X)k kH(X − b 2 /4N0 b =E Q √ ≤ E exp −kH(X − X)k P (X → X) 2N0 Since
b 2 = Tr H† H(X − X)(X b b † kH(X − X)k − X)
(5)
(6)
application of a theorem in [17] yields h i−r b ≤ det It + (X − X)(X b b † /4rN0 P (X → X) − X)
(7)
which is the formula derived differently in [37]. By writing the RHS of this inequality in terms of the product of the b b † , we have eigenvalues λj of the matrix (X − X)(X − X) −r ρ γ −rρ Y b ≤ P (X → X) λj (8) 4 j=1
where γ is the signal-to-noise ratio, and ρ the number of nonzero eigenvalues. From this expression we see that the b b † across all − X) total diversity order of the coded system is rρmin , where ρmin is the minimum rank of (X − X)(X b (“diversity gain”). Moreover, the pairwise error probability depends on the power r of the product of possible pairs X, X b b † . This does not depend on the SNR γ, and displaces the error probability curve rather eigenvalues of (X − X)(X − X) than changing its slope. This is called the “coding gain.” In [3], the pairwise error probabilities obtained from ML interface were compared with those derived from two types of suboptimum receiver interfaces (i.e., of branch metrics): the linear zero-forcing interface and the linear minimummean-square-error interface. It was proved that for large r and finite t these three interfaces give rise to the same pairwise error probabilities, and that these depend on the Euclidean distances between code words: this is the same result as for a transmission occurring over a number of independent, parallel non-fading additive white Gaussian noise (AWGN) channels. Thus, for large r space-time codes can be optimized by using the Euclidean-distance criterion valid for AWGN channel. However, when r → ∞ and t → ∞ while α , t/r ≈ 1, use of a suboptimum interface impairs considerably the error performance of the code. Simulations validate these theoretical findings, and show how asymptotic results can be substantially valid even when the number of antennas is relatively small. Space-time codes when CSI is not available. In a rapidly-changing mobile environment, or when long training sequences are not allowed, the assumption of perfect CSI at the receiver may not be valid. In the absence of CSI at the receiver, [21, 31] advocate unitary space-time modulation, a technique which circumvents the use of training symbols (which for maximum throughput should occupy half of the transmission interval, as seen before). Here the information is carried on the subspace that is spanned by orthonormal signals that are sent to the transmit antennas. This subspace survives multiplication by the unknown channel-gain matrix. A scheme based on differential unitary space-time signals is advocated in [22]. High-rate constellations with excellent performance, designed via algebraic techniques, are described in [19].
Bit-interleaved space-time codes. For single-antenna systems, a coding scheme, called bit-interleaved coded modulation (BICM) [9] was shown to be robust to the channel model, as it provides good performance over the Rayleigh fading channel as well as over the additive white Gaussian noise channel: in fact, it yields a large Hamming distance while at the same time approaching the capacity of high-SNR AWGN channels. As BICM separates the selection of an encoder from the selection of a modulation scheme, the design task is simpler than with “standard” space-time codes. Thus, it seems sensible to investigate how BICM performs over a multiple-antenna channel. Under the block-fading channel model, it may seem at first that BICM cannot buy much. In fact, it operates by transforming a channel into parallel binary channels, and it is known that over block-fading channels binary transmission is less efficient. On the other hand, recent results [25, 36] show that BICM could perform considerably well in several contexts, and hence that BICM warrants some analysis aimed at assessing its limiting performance in a multiple-antenna environment. With BICM, the output of a “mother encoder” is cyclically connected to t transmitters, each including a bit interleaver (whose depth is assumed infinite) followed by a modulator and an antenna. The codes seen by the transmit antennas are punctured versions of the mother code. At the receiving front-end of the channel, the bit metrics are computed, restored to their original order by a de-interleaver, and used for decoding. The receiver is assumed to have perfect knowledge of the “channel-state information,” i.e., of the values taken by the channel gains. The informationtheoretic limits of this scheme are investigated in [4].
BLAST architecture ML detection might be too intensive computationally. Specifically, as t and r increase, the complexity of space-time coding with maximum-likelihood detection may become too large. This motivates the design of suboptimal signalprocessing schemes which exhibit low complexity and still achieve a good portion of the spectral efficiency predicted by the theory. One such scheme was proposed by Foschini [13]. This architecture is called BLAST (Bell Laboratories Layered Space Time) and is capable of realizing a significant fraction of the theoretical capacity using only ordinary modulation and coding techniques. The theory was confirmed in experimental demonstrations at Bell Laboratories. We examine “Vertical Zero-Forcing BLAST” here, in the general case r ≥ t. A convenient algebraic description of this algorithm is the following. The channel matrix H is first decomposed as H = QR using the QR factorization, where R is an upper triangular t × t matrix (normalized so that the diagonal elements are positive), and Q is an r × t matrix with Q† Q = I (notice that if r = t then Q becomes a unitary matrix, i.e., Q† Q = QQ† = I). Observe that this factorization implies an ordering of the transmit antennas, which can be performed in t! ways: this corresponds to the “ordering” part of BLAST, that we shall discuss later on. The receiver uses the “feedforward filter” matrix Q to obtain e Y , Q† |{z} Y |{z} |{z} t×N
t×r r×N †
= Q (QRX + Z) e = |{z} R |{z} X +Z
(9)
t×t t×N
or, explicitly,
e1 y e2 y e3 y .. . et y
=
[R]1,1 [R]1,2 [R]1,3 · · · 0 [R]2,2 [R]2,3 · · · 0 0 [R]3,3 · · · .. .
[R]1,t [R]2,t [R]3,t
···
[R]t,t
0
0
0
x1 x2 x3 .. .
+
xt
e z1 e z2 e z3 .. .
(10)
e zt
e is a t × N matrix whose elements have the same distribution as those of Z. If interface processing were stopped where Z at this stage (i.e., no cancellation took place), the metric would be equivalent to ML: e − RXk2 = kQ† Y − Q† HXk2 = kY − HXk2 kY
(11)
Further processing (the cancellation step) by the ZF-BLAST interface is done by the (nonlinear) feedback filter. This removes the remaining spatial interference resulting from the off-diagonal terms of R, which is achieved by decoding the
subcode transmitted by antenna t first, then subtracting its decoded values from the signal received from antenna t − 1, bt of xt by decoding [R]t,t xt + z ˜t . Next we obtain x bt−1 by decoding and so on. Specifically, we first obtain the estimate x bt + z ˜t−1 , etc. The statistics of [R]i,i depend on the value of i: in particular, the expected value [R]t−1,t−1 xt−1 + [R]t−1,t x of [R]2ii decreases as i increases, which means that he first decoding steps are more at risk of entailing error propagation to subsequent steps. To avoid this error propagation, a possible strategy consists of choosing the ordering of rows of H which is most favorable [15]. BLAST with per-survivor processing. The layered space-time architecture described above is attractive, but exhibits some downsides. First, it requires several independent encoder/decoder pairs running in parallel. Moreover, if used with component trellis codes, not all symbols are decoded with the same decoding delay: this may pose a problem for the underlying Viterbi algorithm. Finally, the interference cancellation procedure on which BLAST is based is prone to error propagation. In order to solve these problems, while keeping the complexity limited, a modification of BLAST was proposed in [6, 7]. There, a single trellis encoder is “wrapped” along the transmit antennas by a diagonal interleaver, and the detection scheme is integrated into a per-survivor processing (PSP) receiver. Thanks to the PSP, the impact of unreliable decisions is greatly reduced.
Conclusions We have highlighted some recent results on multiple-antenna transmission systems. Specifically, we have summarized theoretical performance limits of these systems under the assumptions that the channel gains are known by the receiver alone, by the transmitter and the receiver, or by neither. The design of coding and signal-processing schemes that approach these limits was also addressed.
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