Rayleigh Fading Multi-Antenna Channels

Information theoretic properties of flat fading channels with multiple antennas are investigated. Perfect channel knowledge at the receiver is assumed. Expressions for maximum information rates and outage probabilities are derived. The advantages of transmitter channel knowledge are determined and a critical threshold is found beyond which such channel knowledge gains very little. Asymptotic expressions for the error exponent are found. For the case of transmit diversity closed form expressions for the error exponent and cutoff rate are given. The use of orthogonal modulating signals is shown to be asymptotically optimal in terms of information rate.


Introduction
Spatial diversity has been proposed for support of very high rate data users within third generation wideband Code-Division Multiple Access (CDMA) systems such as cdma2000 [1]. Using multiple antennas, these systems achieve gains in link quality and therefore capacity. Classically, diversity has been exploited through the use of either beam-steering (for antenna arrays with correlated elements), or through diversity combining (for independent antenna arrays). Recently it has been realized that co-ordinated use of diversity can be achieved through the use of space-time codes [2]. Increasing the number of antennas and using space-time codes involves both a physical and computational complexity/performance trade-off. Simple schemes such as multi-carrier transmit diversity or array processing techniques, such as maximum ratio combining may be easily implemented. How is their performance limited, when compared to powerful space-time codes?
In this paper we attempt to answer some of these questions from an information theoretic point of view. We derive a new expression for the capacity of a Rayleigh fading channel with multiple antennas. We investigate the error exponent for such channels and find an asymptotic expression for large systems. We also specifically consider transmit diversity. We derive a closed form expression for the error exponent This work was supported by Nortel Networks. and show that the cut-off rate increases logarithmically with the number of antennas. Although capacity is near to maximum for a small number of antennas (typically 2 or 3), use of additional antennas improves performance through increase of the error exponent. We also show that asymptotically use of orthogonal carriers for each transmit antenna is optimal.

Channel Model
Let us consider a point-to-point channel with t transmit antennas and r receive antennas. The information theoretic analyses presented so far in the literature have concentrated on the flat fading case, we likewise restrict our present scope. Associated with each transmit/receive pair is a possibly time varying complex scalar channel gain. At each symbol interval l = 1, 2, . . . , L, the received matched-filtered vector y[l] ∈ C r depends on the transmitted vector, x[l] ∈ C t according to Element y j [l] is matched-filter output j, while x i [l] is the transmit signal at input i. The matrix H[l] ∈ C r×t has as elements H ji [l] ∈ C, which are the complex channel gains (representing flat Rayleigh fading) between input i and output j at time l. The vector n[l] contains i.i.d. circularly symmetric Gaussian noise samples, E [n[l]n * [l]] = σ 2 I r . We place the following power constraint on the transmitted signal (independent of t), We denote the signal to noise ratio (SNR) as γ = P/σ 2 .
The channel matrix may either be fixed, or time varying, and throughout the paper, we assume that it is known at the receiver (but not at the transmitter. Rayleigh fading will be modelled as follows.
Telatar [3] gives the capacity in the cases when H is known at the receiver and either known or unknown at the transmitter. The case where the channel matrix is completely unknown is not yet solved. Early work can be found in [4,5].

Capacity
Telatar [3] has found the capacity of the Rayleigh space-time channel when H is chosen independently each symbol interval and is known at the receiver. The following expression is a simplification of that result. Theorem 1. At each symbol interval, let H be selected according to Definition 1. Let m = min{r, t} and n = max{r, t}. Then the capacity of the channel (1) is given by Where L α k is the generalized Laguerre polynomial of order k [6]. Capacity is achieved for x circularly symmetric zero mean complex Gaussian vector with E [xx * ] = I t P/t.
The expression (2) involves the sum of only two integrals, compared to the m-fold sum required by [3]. We can also prove the following simple bounds.

Error Exponent
It is also interesting to consider the error exponent [7] for space-time channels. The error exponent gives some indication of how difficult it may be to achieve a certain bit error rate. Telatar [3] gives the following lower bound to the error exponent. Lemma 1. The probability of error averaged over all randomly selected (N, N R) block codes is bounded Reliability functions for fading channels with correlated multiple antennas have also been considered in [8] for t = 1, see also [9].
We can prove following limiting theorem using the theory of random determinants.
Theorem 3. Let t/r be fixed. The following limiting relation holds.

Transmit Diversity
In this section, we shall consider transmit diversity, r = 1, t > 1. It is easy to show that as t → ∞, C Rayleigh → log(1 + γ) , which is as if all the power were transmitted over a single, non-faded link. By increasing the number of antennas, the effects of the fading may be completely removed. Increasing the number of transmit antennas also has the desirable effect of increasing the error exponent, as we shall now show.

(7)
Where Ψ is the confluent hypergeometric function [6, §9.2]. Furthermore, letting C g (γ) = log(1 + γ) be the Gaussian channel capacity with SNR γ we have the following limiting expression for the error exponent where for the second case, C g γ 2 − γ 2(γ+2) < R ≤ C g (γ) we have a parametric representation via 0 ≤ ρ ≤ 1 and Figure 1 compares the error exponents for the additive white Gaussian noise channel [7, (7.4.33)] and the limiting t → ∞ transmit diversity channel with Rayleigh fading (8). The signal to noise ratio is 0dB. Further investigations have shown that the gap in the error exponents only disappears as γ → 0. It is interesting to note that although t → ∞ results in C Rayleigh → C g (γ) there is an inherent complexity penalty to be paid (at least in terms of the error exponent).  Closed form determination of E r for finite t using (7) seems difficult, although numerical methods can be used. We shall therefore concentrate on the cut-off rate, R 0 = E 0 (1), which gives the error exponent up the the critical rate, and a lower bound beyond that. Substituting ρ = 1 into the expressions of Theorem 4 we obtain the following corollary.
For t > 1 the cut-off rate may be lower bounded as follows, This bound is tight as t → ∞ or γ → ∞. Hence R 0 increases at least as the logarithm of t/(t − 1). Figure 2 shows R 0 (solid line) and the lower bound (10) (dashed) plotted versus t for various SNR.

Orthogonal Transmit Diversity
It is interesting to consider the use of mutually orthogonal transmit waveforms for each transmit antenna, as proposed for cdma2000 [1, §3.2.1.1.5]. We shall continue to consider transmit diversity only (r = 1), and assume codebooks with i.i.d. circularly symmetric complex Gaussian letters. If the channel is according to Definition 1, the transmission rate is upper bounded by C Rayleigh as given by Equation (2).
It is easy to see (e.g. through application of the data processing theorem [11] that restriction to orthogonal carriers can only decrease the maximum possible mutual information, when compared to the use of unconstrained space-time codes. In this section we will show that use of a sufficiently large number of transmit antennas makes this penalty insignificant. This may be interesting since use of orthogonal carriers should result in simple receiver structures [12,13]. Narula et al [14] found the average mutual information (assuming a Gaussian codebook) in the case of orthogonal signaling. We can prove the following result which gives an expression for E [I ⊥ ] and shows its limiting behavior for large signal-to-noise ratios.
Theorem 5. Let H be randomly selected every symbol interval according to Definition 1, and let H be known at the receiver. Then the maximum transmission rate, under the assumption of orthogonal transmit diversity is given by where Ei(x) = ∞ −x e −x /x dx. Furthermore, as the signal-to-noise ratio increases, the difference between E [I ⊥ ] and log(1 + γ) tends to a small constant, where C ≈ 0.577 is Euler's constant [6]. E [I ⊥ ] → log(1 + γ) in the sense that their ratio tends to unity.
It is easy to verify that for r = 1 and increasing t, C Rayleigh → log(1+γ). Hence even though a rate penalty is incurred for using orthogonal transmit diversity, this penalty disappears as the signal-to-noise ratio and the number of antennas is increased.

Conclusion
In this paper we have given several calculations regarding the capacity and error exponents of multiple antenna Gaussian channels. In particular we have found a limiting expression for E 0 (ρ), and a closed form expression for the case of transmit diversity (r = 1). We further showed that for r = 1 the cut-off rate increases logarithmically with the number of transmit antennas. Although capacity is near to maximum for a small number of antennas (typically 2 or 3), use of additional antennas improves performance through increase of the error exponent. We have also shown that use of a sufficiently large number of transmit antennas offsets the rate penalty suffered due to orthogonal transmissions from each antenna, such as those described for cdma2000.