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MIMOlecture - Capacity of multiple-input...

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15/11/02 Capacity of multiple-input multiple-output (MIMO) systems in wireless communications Bengt Holter Department of Telecommunications Norwegian University of Science and Technology NTNU 1

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Outline 15/11/02 Introduction Channel capacity Single-Input Single-Output (SISO) Single-Input Multiple-Output (SIMO) Multiple-Input Multiple-Output (MIMO) MIMO capacity employing space-time block coding (STBC) Outage capacity SISO SIMO MIMO employing STBC Summary NTNU 2
Introduction 15/11/02 MIMO = Multiple-Input Multiple-Output Initial MIMO papers I. Telatar, ”Capacity of multi-antenna gaussian channels,” AT&T Technical Memorandum, jun. 1995 G. J. Foschini, ”Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas”, Bell Labs Technical Journal, 1996 MIMO systems are used to (dramatically) increase the capacity and quality of a wireless transmission. Increased capacity obtained with spatial multiplexing of transmitted data. Increased quality obtained by using space-time coding at the trans- mitter. NTNU 3

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Entropy 15/11/02 For a discrete random variable X with alphabet X and distributed according to the probability mass function p ( x ), the entropy is defined as H ( X ) = x ∈X log 2 1 p ( x ) · p ( x ) = x ∈X log 2 p ( x ) · p ( x ) = E log 2 1 p ( x ) . (1) The entropy of a random variable is a measure of the uncertainty of the random variable; it is a measure of the amount of information required on the average to describe the random variable. With a base 2 logarithm, entropy is measured in bits. NTNU 4
Gamma distribution 15/11/02 X follows a gamma distribution with shape parameter α > 0 and scale parameter β > 0 when the probability density function (PDF) of X is given by f X ( x ) = x α 1 e x/β β α Γ( α ) . (2) where Γ( · ) is the gamma function (Γ( α ) = 0 e t t α 1 dt [ ( α ) > 0]). The short hand notation X ∼ G ( α, β ) is used to denote that X follows a gamma distribution with shape parameter α and scale parameter β . Mean : µ x = E{ X } = α · β . Variance : σ 2 x = E{ X 2 } − µ 2 x = α · β 2 . NTNU 5

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SISO 15/11/02 SISO = Single-input Single-output Representing the input and and output of a memoryless wireless chan- nel with the random variables X and Y respectively, the channel ca- pacity is defined as C = max p ( x ) I ( X ; Y ) . (3) • I ( X ; Y ) denotes the mutual information between X and Y and it is measure of the amount of information that one random variable contains about another random variable. According to the definition in (3), the mutual information is maximized with respect to all possible transmitter statistical distributions p ( x ). NTNU 6
SISO cont’d 15/11/02 The mutual information between X and Y can also be written as I ( X ; Y ) = H ( Y ) H ( Y | X ) . (4) From the equation above, it can be seen that mutual information can be described as the reduction in the uncertainty of one random variable due to the knowledge of the other.

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