# 100309-lecture9 - EE522 Communications Theory Spring 2010...

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1 EE522 Communications Theory Spring 2010 Instructor: Hwang Soo Lee Lecture #9 – Optimal Receiver Design

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2 Announcements ± Project Proposals ± HW #3 Due Thursday, March 11 ± Handout ² Course Notes #9
3 Modulation ± We want to modulate digital data using signal sets which are: ² bandwidth efficient ² energy efficient ± A signal space representation is a convenient form for viewing modulation which allows us to: ² design energy and bandwidth efficient signal constellations ² determine the form of the optimal receiver for a given constellation ² evaluate the performance of a modulation type

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4 Problem Statement ± We transmit a signal , where is nonzero only on . ± Let the various signals be transmitted with probability: ± The received signal is corrupted by noise: ± Given , the receiver forms an estimate of the signal with the goal of minimizing symbol error probability . ) ( t s ) ( ) ( ) ( t n t s t r + = ) ( t s ) ( t r ) ( ˆ t s
5 Noise Model ± The signal is corrupted by Additive White Gaussian Noise (AWGN) . ± The noise has autocorrelation and power spectral density . ± Any linear function of will be a Gaussian random variable. ) ( t n ) ( t n ) ( t n

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6 Signal Space Representation ± The transmitted signal can be represented as: ,where . ± The noise can be represented as: where , and .
7 Signal Space Representation (continued) ± The received signal can be represented as: where .

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8 The Orthogonal Noise: ± The noise can be disregarded by the receiver:
9 We can reduce the decision to a finite dimensional space!

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100309-lecture9 - EE522 Communications Theory Spring 2010...

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