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A brief review of Shannon's Information Theory
The shannon capacity of a general radio channel with transmission power \rho = P/n_T and noise power n is given by
C = E\big(\log \det(I_{N_R} + \frac{\rho}{n}HH^T) \big)
where I is the identity matrix and H is the channel transform matrix as described above. Since this holds for a gaussian channel, we assume that the transmission vector x is a set of gaussian distributed, circularly symmetric, independent symbols.
If we fix n_R_ and increase n_T, the strong law of large numbers dictates
\det(I_{N_R} + \frac{\rho}{n}HH^T)
will tend towards
n_R \log(1+\rho). For an exact solution, we can, noting that
HH^* is a non-negative definite matrix, replace it by its eigenvalues (by the spectral decomposition theorem). The capacity thus becomes equal to
C = E \sum_i \big(\log (N_R + \frac{\rho}{n}\lambda_i) \big)
, where \lambda_i is the ith eigen value of HH^*, once again confirming our original assertion that it is the eigenvalues of the channel transform matrix which determines the final SNR of the signal.
The eigenvalues of a wishart Matrix
The eigenvalues of a wishart matrix obey the wishart distribution. By taking the expectation and solving it, the exact capacity was shown by [Elatar], [Foschini] linear to min(n_R,n_T); in other words, by increasing the number of antennae, we get a linear increase in capacity. Compare this to increase in the value of \rho in the first equation, which simply gives a logarithmic increase in capacity, we can understand the improvements brought about by using spatial diversity.
The role of the transmitter
Uptil now, we haven't considered the role of the transmitters or the inter-relations between the transmitted symbols as a part of this problem, treating the transmitted symbols as random variables. As it happens, the transmitter can play a significant role in achieving the promised capacity improvements of multiple antenna systems, by controlling the inter-relationships between the transmitted symbols and the power level of the symbols. This is called pre-coding. The ability of the transmitter to optimally pre-code the transmitted symbols depends on the amount of knowledge that it has of the channel.
Let the inter-relations between the different transmitted symbols be described by an n_T by n_T matrix Q, known as the covariance matrix. The channel transfer equation thus becomes R = \rho I + HQH^*. If we do a singular value decomposition of HQH^ = \Lambda^{\frac{1}{2}}UQU^*\Lambda^{\frac{1}{2}} = \Lambda^{\frac{1}{2}}\tilde{Q}\Lambda^{\frac{1}{2}} where Q and \tilde{Q} are statistically identical and \Lambda contains the eigenvalues of HH^*.
Optimal pre-coding
[Telatar] and others have shown that
\det(I_r + \Lambda^{\frac{1}{2}}\tilde{Q}\Lambda^{\frac{1}{2}})
\le \Pi(1+\tilde{Q}_{ii}\lambda_i)
where
\lambda_i is the ith eigenvalue of
HH^*. The optimal solution is achieved when
\tilde{Q} is a diagonal matrix with the ith diagonal entry given by
Q_{ii} = \mu - \lambda_i^{-1} .
If the transmitter is aware of the channel matrix H, it can compute the eigenvalues of HH^* and then the optimal Q using the water filling algorithm. However in many situations, it is not possible for the transmitter to have full channel knowledge, even though it has statistical knowledge of the channel. In other cases, it is not feasible to run the water-filling algorithm for every iteration, especially in fast-fading channels. Simplified versions of the water-filling algorithm such as iterative water filling, etc. have been evolved, most of which are asymptotically optimal. In the simplest case, the "constant water filling algorithm" can be used, where the power is simply equally distributed over all transmitting entities. Cioffi, et al, [Ciofficonstant] have shown that performance of the constant power water filling algorithm can come very close to that of the optimal algorithm.
[Ciofficonstant] Yu, Cioffi, "On Constant Power Water-filling", ICC 2001
[Telatar] I. Emre Telatar, "Capacity of Multi-Antenna Gaussian Channels"
[Foschini] Foschini, Gans, "On the Limits of Wireless communications in a fading environment when using multiple antennae"
[Paulraj] D.Gore, R.W. Heath, A. Paulraj, "On the Performance of the Zero Forcing Receiver in Presence of Transmit Correlation"
[Edelman] Edelman, "Eigenvalues and the condition number of random matrices"
[Winters] Winters, "On the capacity of radio communication systems with diversity in a rayleigh fading environment" IEEE JSAC, June 1987
[V-Blast]Wolniasky, Foschini, Golden, Valenzuela, "V-BLAST: An Architecture for Realizing Very High Data Rates Over the Rich-Scattering Wireless Channel"
[Giannakis1] Giannakis, Zhou, "Optimal Transmit-Diversity Precoders for Random Fading Channels"
[Alamouti] Sivash Alamouti "A Simple Transmit Diversity Technique for Wireless Communications", IEEE JSAC, VOL. 16, NO. 8, OCTOBER 1998
In the next section we shall discuss space time coding techniques.
Categories: Wireless
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