Lecture 12 - ECE 614: Principles of Digital Communications,...

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Unformatted text preview: ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Lecture 12: February 25, 2010 Outline: Review of probability Probabilistic Detection 1 ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 R( )=E[X * (t)X(t+ )] and =E[X(t)] for WSS process Key understanding of a WSS random process , , , t x t x t X T T T WSS X dt t X t X E T t X t X E R * * 2 1 lim 1 i i x T T T R E dt t X t X T E * 2 1 lim T dt t x t x R * 1 lim where T i i T x T i 2 i x T T T T T T WSS E dt t X T E dt t X E T t X E 2 1 lim 2 1 lim T T i T x dt t x T i 2 1 lim where 2 ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins ECE 614: Principles of Digital Communications, Colorado State University, Fort Collins Spri Spring 2010 ng 2010 Properties R( )=E[X * (t)X(t+ )] (For a WSS random process) t X E R X power . 1 2 , of power the is i i x X i x R E R t x R R R X X symmetry Hermitian 2. * i i i i x x x x R E R E R R * R R t X t X E R R X X X X with Consider : Proof 0. lag at magnitude max 3. * 2 i i i i x x x x R E R E R R R m R R R t X t X E then If . 4 then equality, If 2 j X X e t X t X R R jm j e t X m t X e t X t X F R F X X X X X ansform) Fourier tr the denotes ( s frequencie all for . 5 , R F R F E R E F 3 t X R X of pdf about the nothing says 6....
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Lecture 12 - ECE 614: Principles of Digital Communications,...

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