100302-lecture7 - EE522 Communications Theory Spring 2010...

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1 EE522 Communications Theory Spring 2010 Instructor: Hwang Soo Lee Lecture #7 – Review of Vector Spaces/Gram- Schmidt Procedure for Obtaining Basis Functions
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2 Announcements ± HW #2 Due February 25 ± Project Proposals Due March 9 ± Handouts: ² Lecture #7
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3 Block Diagram of Digital Communications System
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4 Example of Signal Constellation Diagram: QAM ±
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5 A New Way of Viewing Modulation ± The I/Q representation of modulation is very convenient for some modulation types. ± We will examine an even more general way of looking at modulation using signal spaces. ± By choosing an appropriate set of axis for our signal constellation, we will be able to: ² Design modulation types which have desirable properties. ² Construct optimal receivers for a given type of modulation. ² Analyze the performance of modulation types using very general techniques.
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6 Vector Spaces ± An n-dimensional vector consists of n scalar components ± The norm (length) of a vector is given by: ± The inner product of two vectors ± and is given by: v = = n i i v 1 2 || || v
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Basis Vectors ± A vector may be expressed as a linear combination of it’s basis vectors : where . ± Think of the basis vectors as a coordinate
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100302-lecture7 - EE522 Communications Theory Spring 2010...

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