MATHEMATICAL MODEL OF THE MIMO LINK

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In 1, we have reviewed the continuous-time models for at- and frequency-selective fading channels. We also illustrated how to discretize a continuous-time channel model to the equivalent discrete-time channel model for at fading and frequency-selective fading channels. In this section, we shall generalize the concept of discrete-time fading channels to a probabilistic channel with states. For notation convenience, uppercase (capital) X denotes random variable, while lowercase x denotes a realization of the random variable. Bold X denotes a vector or matrix of random variables, while normal X denotes a scalar.

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MATHEMATICAL MODEL OF THE MIMO LINK

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Probabilistic Channels with States

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The MIMO transmitter produces a channel input symbol X X, which is an (nT 1)-dimensional vector over the complex eld, per channel use. The corresponding channel output symbol is given by Y Y, which is an nRdimensional vector over the complex eld. The probabilistic MIMO channel can be characterized by the (nR nT)-dimensional channel state H H, which is again over the complex eld. In general, each channel state realization H = h speci es a channel transition probability p(y|x, h) (if X and Y are discrete sets) or f(y|x, h) (if X and Y are continuous sets). When X and Y are all nite sets, the channel is called discrete channel with states. For instance, given any channel state realization h, the probability of receiving y given x is transmitted is given by p(y|x, h). On the other hand, when X and Y are all continuous sets, the channel is called a continuous channel with states. Given any channel state realization h, the conditional channel transition probability becomes f(y|x, h), where the conditional probability of receiving Y [y, y + dy] given x is transmitted and the current channel state h is given by f(y|x, h)dy. Without loss of generality, we shall assume discrete inputs and discrete outputs for simplicity unless otherwise speci ed. Generalization to continuous inputs and continuous outputs follows the standard argument as in Reference 44. The channel transition probability of MIMO channels can be completely characterized by the conditional channel transition probability and the channel state sequence probability as

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N N N N N N p(y 1 x1 ) = p(y 1 x1 , h 1 ) p(h 1 )

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N h1

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N N where x1 = [x1, . . . , xN] denotes a block of N transmit symbols, y1 = [y1, . . . , N yN] denotes a block of N received symbols, and h1 = [h1, . . . , hN] denotes a block of N channel states. In general, the output symbol yn at time n depends not only on the current transmitted symbol xn but also on the past and future transmitted symbols. If that s the case, the probabilistic channel is said to have memory. Otherwise, the probabilistic channel is said to be memoryless.

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De nition 2.1 (Memoryless Channels) A channel is called memoryless if the unconditional transition probability can be expressed into a product form:2

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N N p(y 1 x1 ) = n =1 p(y n x n ) N

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(2.1)

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We look at some important examples below.

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Please refer to 1 for a more general de nition of memoryless channe [100]. However, throughout the book, we assume the feedback channel carries channel state only and hence, the two de nitions are the same.

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MIMO LINK WITH PERFECT CHANNEL STATE INFORMATION

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Fast Flat Fading MIMO Channels. For example, the fast at fading MIMO channel introduced in 1 is an example of memoryless continuousinput continuous-output MIMO channel because the channel transition probability can be decomposed into a product form. The discrete-time equivalent channel model is given by y n = h n xn + zn (2.2)

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where zn is an (nR 1)-dimensional i.i.d. complex Gaussian random vector sequence with covariance h0InR. In the model, {xn} are the complex-valued nT 1 channel input, {yn} are the complex-valued nR 1 channel output, and {hn} are the nR nT i.i.d. complex Gaussian channel state sequence.3 Given {hn} and {xn}, the channel transition probability is given by p(y 1 , . . . , y N x1 , . . . , x N , h 1 , . . . , h N ) 1 -1 exp - (y n - h n x n ) * y (y n - h n x n ) = N n (p y ) =

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