Stanford EE 102 - Transfer Functions and Convolution (19 pages)

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Transfer Functions and Convolution



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Transfer Functions and Convolution

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Pages:
19
School:
Stanford University
Course:
Ee 102 - Introduction to Signals and Systems

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EE 102 spring 2001 2002 Handout 15 Lecture 8 Transfer functions and convolution convolution transfer functions properties examples interpretation of convolution representation of linear time invariant systems 8 1 Convolution systems convolution system with input u u t 0 t 0 and output y y t t 0 h u t d t 0 h t u d abbreviated y h u in the frequency domain Y s H s U s H is called the transfer function TF of the system h is called the impulse response of the system block diagram notation s u h Transfer functions and convolution y u H y 8 2 Properties 1 convolution systems are linear for all signals u1 u2 and all R h u1 u2 h u1 h u2 2 convolution systems are causal the output y t at time t depends only on past inputs u 0 t 3 convolution systems are time invariant if we shift the input signal u over T 0 i e apply the input 0 t T u t u t T t 0 to the system the output is y t 0 t T y t T t 0 in other words convolution systems commute with delay Transfer functions and convolution 8 3 4 composition of convolution systems corresponds to multiplication of transfer functions convolution of impulse responses composition u u A B BA y y ramifications can manipulate block diagrams with transfer functions as if they were simple gains convolution systems commute with each other Transfer functions and convolution 8 4 Example feedback connection u y G in time domain we have complicated integral equation y t t 0 g t u y d which is not easy to understand or solve in frequency domain we have Y G U Y solve for Y to get Y s H s U s H s G s 1 G s as if G were a simple scaling system Transfer functions and convolution 8 5 General examples first order LCCODE y y u y 0 0 take Laplace transform to get 1 U s Y s s 1 transfer function is 1 s 1 impulse response is e t t integrator y t u d 0 transfer function is 1 s impulse response is 1 delay with T 0 y t 0 t T u t T t T impulse response is t T transfer function is e sT Transfer functions and convolution 8 6 Vehicle suspension system simple model



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