SPFirst-L20

SPFirst-L20 - Signal Processing First READING ASSIGNMENTS...

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11/3/2003 © 2003, JH McClellan & RW Schafer 1 Signal Processing First Lecture 20 Convolution (Continuous-Time) 11/3/2003 © 2003, JH McClellan & RW Schafer 3 READING ASSIGNMENTS ± This Lecture: ± Chapter 9, Sects. 9-6, 9-7, and 9-8 ± Other Reading: ± Recitation: Ch. 9, all ± Next Lecture: Start reading Chapter 10 11/3/2003 © 2003, JH McClellan & RW Schafer 4 LECTURE OBJECTIVES ± Review of C-T LTI systems ± Evaluating convolutions ± Examples ± Impulses ± LTI Systems ± Stability and causality ± Cascade and parallel connections 11/3/2003 © 2003, JH McClellan & RW Schafer 5 Linear and Time-Invariant (LTI) Systems ± If a continuous-time system is both linear and time-invariant, then the output y(t) is related to the input x(t) by a convolution integral convolution integral where h(t) is the impulse response of the system. = = ) ( ) ( ) ( ) ( ) ( t h t x d t h x t y τ
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11/3/2003 © 2003, JH McClellan & RW Schafer 6 Testing for Linearity x 1 ( t ) x 2 ( t ) y 1 ( t ) y 2 ( t ) w ( t ) y ( t ) x ( t ) x 2 ( t ) x 1 ( t ) w ( t ) y ( t ) 11/3/2003 © 2003, JH McClellan & RW Schafer 7 Testing Time-Invariance x ( t ) x ( t t 0 ) y ( t ) w ( t ) y ( t t 0 ) t 0 w ( t ) y ( t t 0 ) t 0 11/3/2003 © 2003, JH McClellan & RW Schafer 8 Ideal Delay: ± Linear ± and Time-Invariant ) ( ) ( d t t x t y = ) ) (( ) ( 0 t t t x t w d = ) ) (( ) ( 0 0 d t t t x t t y = ) ( ) ( ) ( ) ( 2 1 2 1 t by t ay t t bx t t ax d d + = + 11/3/2003 © 2003, JH McClellan & RW Schafer 9 0 0
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SPFirst-L20 - Signal Processing First READING ASSIGNMENTS...

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