### MIMO Channel Model

Consider a point-to-point wireless communication channel equipped with transmit and receive antennas. This channel can be modeled by a multiple-input multiple-output (MIMO) system, which can be represented as a baseband channel matrix **H** of size . The entry of **H** at the *j*-th row and *i*-th column, , represents the complex channel coefficient relating to the *i*-th transmit antenna and the *j*-th receive antenna. We assume that the antennas are placed at an enough distance between each other, such that the channel link between each transmit-receive antenna pair experiences independent fading. The fading is modeled as flat (non-frequency selective), since frequency selective fading can be combated by the orthogonal frequency division multiplexing (OFDM) technique.

The fading distribution is assumed to be Rayleigh, or equivalently, complex Gaussian, which represents the worst case fading situation. Proper normalization is assumed such that each entry can be modeled as a complex Gaussian random process with zero mean and unity variance (variance 0.5 for both real and imaginary parts). We denote this as , where denotes complex Gaussian distribution with mean and variance .

The input to the MIMO channel is represented by an vector , where *x*_{i} denotes the transmitted (modulated) symbol from the *i*-th transmit antenna. The output from the MIMO channel is represented by an vector , where *y*_{j} denotes the received symbol from the *j*-th receive antenna. The additive white Gaussian noise (AWGN) is represented by an vector , where *n*_{j} denotes the equivalent complex noise at the *j*-th receive antenna. The channel equation can be written as

.

*(1)*Graphically, this can be illustrated by Figure 1.

For convenience, we define

.

We apply a singular value decomposition (SVD) to the matrix **H** and get

* (2)* where

**D** is an

diagonal matrix. Its diagonal entries are the singular values of

**H**, that is, the non-negative square roots of the eigenvalues of

**HH**^{H}. We denote the eigenvalues of

**HH**^{H} as

, thus

and

are

and

unitary matrices with left and right singular vectors of

**H** as their columns, respectively.

and

are also called the eigenvectors of

**HH**^{H} and

**H**^{H}**H** associated with

, respectively. The eigenvalues and eigenvectors are related as

,

.

Substituting Eq. (2) into (1), we obtain

* (3)* where

.

Eq. (3) readily shows a parallel channel model. To put it more explicitly, we can rewrite Eq. (3) in a component-wise form

,

*(4)*,

and

.

*(5)*
Eq. (4) and (5) are further illustrated in Figure 2.

*Figure 2. *Parallel equivalent subchannel model

This clearly shows that the MIMO channel has been converted into *m* parallel subchannels. The equivalent channel input and output are and , respectively. The subchannel power gains are .