Lect4 - Stochastic Process 10/13/2006 Lecture 4...

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Unformatted text preview: Stochastic Process 10/13/2006 Lecture 4 Fundamentals of Detection NCTUEE Summary In this lecture, I will discuss: • Complex Gaussian Random Vector • Circularly Symmetric Complex Gaussian and Its Density • Fundamental Concepts of Detection Notation We will use the following notation rules, unless otherwise noted, to represent symbols during this course. • Boldface upper case letter to represent MATRIX • Boldface lower case letter to represent vector • Superscript ( · ) T and ( · ) H to denote transpose and hermitian (conjugate transpose), respectively • Upper case italic letter to represent RANDOM VARIABLE 4-1 1 Complex Gaussian Vector 1. A complex Gaussian vector x = x r + j x i is a random vector whose real part x r and imaginary part x i are collectively jointly Gaussian. (1) The joint pdf of an n-dimensional complex random vector x = x r + j x i is defined to be the joint pdf of the 2 n-dimensional real random vector [ x T r , x T i ] T . (2) A complex Gaussian vector is completely specified by the mean m x = E [ x ], the covariance matrix K x = E [( x- m x )( x- m x ) H ] , and the pseudo-covariance matrix J x = E [( x- m x )( x- m x ) T ] of the complex vector x . 2. In applications of wireless communication, we are almost exclusively interested in complex random vectors that have the circular symmetry property: x is circular symmetric if e jθ x has the same distribution as x for any θ . (1) For a circular symmetric complex random vector x , its mean vec- tor is a zero vector and its pseudo-covariance matrix is a zero matrix. (2) A circular symmetric complex Gaussian random vector x is com- pletely specified by its covariance matrix and is denoted as x ∼ CN (0 , K x ) . 4-2 3. A complex Gaussian random variable X = X r + jX i with i.i.d. zero mean Gaussian real and imaginary components is circular symmetric. (1) The statistics of X are fully specified by the variance σ 2 = E [ | X | 2 ] , denoted as X ∼ CN (0 ,σ 2 ) . (2) We can represent X in polar form X = || X || e j Θ , where the phase Θ is uniform over [0 , 2 π ] and independent of the magnitude || X || , which has a density given by f || X || ( r ) = r σ 2 exp- r 2 2 σ 2 ¶ , r ≥ and is known as a Rayleigh random variable. 4-3 4. A collection of n i.i.d. CN (0 , 1) random variables forms a standard cir- cular symmetric Gaussian random vector w and is denoted by CN (0 , I )....
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Lect4 - Stochastic Process 10/13/2006 Lecture 4...

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