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General theory of operation
Instead of giving a very mathematical treatment (the interested reader is pointed to excellent articles by Foschini, Elatar, Chiani and Edelman, see references below), we will try to give an intuitive understanding of the underlying theory, which allows us to get such amazing results.
The system model is as follows:
We have n_T_ transmitters, and n_R_ receivers. Typically, though not necessarily, the two are matched. The transmitters transmit a signal into the air. The transmitted signal is called the signal vector; we denote it here by x_n_, a complex vector of size n_T_, each of whose entries are a complex number, referring to a specific symbol in the modulation constellation. Since there is a total transmission power N, the sum of the transmission powers of the individual transmit antennae has to be less than N. When a signal x is transmitted by the ith transmitter, and is received by the jth receiver, it is transmuted by the path conditions between this pair; this path condition is represented by a complex multiplier h_ij_, which may change both the magnitude and the phase of the transmitted signals. The n_T_xn_r_ matrix H is the matrix of the path multiplers for the entire system, whose ij^th^ entry represents the path condition between the i^th^ transmitter and the j^th^ receiver. We assume that the conditions are time-invariant, at least for short durations of time longer than a burst transmission period.
The received signal then becomes y = xH + n where n is the additive noise vector.
A second diversion on Hermitian matrices
H is a complex matrix. H^^ is the conjugate transpose of H. For example, if H is given by 
, the conjugate transpose of H is given by 
. The product of the two matrices K=HH^^ is a Hermitian matrix and has very useful properties. Specifically, it can be decomposed into real eigenvalues, and a set of eigenvectors which are orthonormal to each other and thus can be used to form the basis of a vector space
It is possible for a node to get full knowledge of the channel by measuring the matrix H directly; this can be done by transmitting pilot tones of standard power and contents and using them to estimate the transfer function.
The receiving method - an example
To illustrate how the mechanism can work theoretically, we take a simple scenario.
In this scenario, both the transmitter and the receiver have full knowledge of channel, i.e. they know the channel transfer matrix H. This is possible in a TDD environment, where both systems are alternately transmitter and receiver and are transmitting pilot tones to each other on the same frequency band. To transmit a signal a, the transmitter chooses a left eigenvector of HH^^, x_i_ and uses it to modulate a. The signal received at the receiver is thus:
y_i_ = ax_i_H + N.
Theoretically, the receiver can recover the original signal by the following steps.
- . The received signal is multiplied by H^^, the conjugate transform of H, to get:
y'_i_ = ax_i_HH^^ + NH^^ = a_i_x_i_+ NH^^.
The second equation comes from the fact that _i_ is the eigenvalue corresponding to x_i_ and the properties of eigenvectors.
- . We then multiply it by the transpose of the original eigenvector x^T^_i_ to get:
y''_i_ = a_i_ + NH^^x^T^_i_.
Since the eigenvectors are orthonormal, we get the original transmitted signal a, scaled by the eigenvalue i, plus a noise term. If the eigenvalue is large enough, the SNR is good and we get a good received signal.
In reality, the reciever does not directly multiply with H^^, but uses iterative equalization? techniques to extract the received signal. The effect remains the same - the optimal linear filter at the receiver is given by w = R^-1^x^T^_i_ [Winters84], where R is the interference plus noise correlation matrix and is given by R = I + HH^^. Adding the scaled identity vector to HH^^ does not, obviously, have any impact on the eigenvalues of HH^^. There are a vast number of receiver structures which provide optimal decoding of the transmitted signal, given the knowledge of H. However, as this example shows, the SNR of the final decoded signal is inextricably tied to the magnitude of the eigenvalues of the matrix K. We shall discuss this issue subsequently.
The channel matrix
The eigenvalues of K depend on the channel matrix, so let us spend some time looking at the channel transform function H. It is obvious that H is very much dependent on the nature of the physical channel. For example, if the physical media is such that there are exactly n transmitters and receivers, and the ith receiver can only receive from the ith transmitter, and there is a fixed attenuation , then H becomes I, where I is the identity matrix.
The case that we are going to discuss in this article is the channel where there is no clear line-of-sight signal, instead there are multiple non-line-of-sight signals of similar strength. Such a channel is termed a Rayleigh channel and the noise distribution follows the Rayleigh distribution. The transfer parameter h_i,j_ for each pair i,j of transceivers have real and imaginary parts which are zero mean, i.i.d. normal variables, with a variance of 1/2. This ensures that modulus of the transfer function is 1. Then the product HH^^ becomes a definite matrix with random non-negative entries which is also known as a Wishart matrix. It is the statistical properties of Wishart matrices from which the improvements of multiple-antennae systems are derived.
Wishart matrices have been studied for a long time in various disciplines of multivariate statistical analysis and statistical mechanics. For example, if we take n samples V = {v_1_, v_2_,v_3_,v_4_,...v_n_} of a process with a multivariate normal distribution i.e. a Brownian distribution in multiple dimensions, then the sample covariance matrix times.
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