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lecture_9

# lecture_9 - 2.160 System Identification Estimation and...

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2.160 System Identification, Estimation, and Learning Lecture Notes No. 9 March 8, 2006 Part 2 Representation and Learning We now move on to the second part of the course, Representation and Learning. You will learn various forms of system representation, including linear and nonlinear systems. We will use “Prediction” form as a generic representation of diverse systems. You will also learn data compression and learning, which are closely related to system representation and prediction. 5 Prediction Modeling of Linear Systems 5.1 Impulse Response and Transfer Operator (Review) Linear Time-Invariant system u(t) ) 1 ( g y(t) ) 2 ( g ) ( g( τ ) y(t) y(t) t g Impulse response: a complete characterization t t t Continuous time impulse response Discrete time impulse response Any linear time-invariant system can be characterized completely with Impulse Response: ( ( ) t g ) . For an arbitrary input, { s s u t } , the output y ( t ) is given by the convolution of the input and the impulse response given by Continuous Time Convolution t y ) = g ( τ ) t u τ τ (1) ( ( ) d 0 For discrete-time systems, an impulse response is given by an infinite time series: 1

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g ( ), 0 g ( ), 1 g ( ), 2 " , or { g ( τ τ = 0 } ) Output { s s y t } { s s u t } ( ) ( )
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