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Convolution

Convolution. 3.1-3.2. An Integrator Circuit. Motivation. How does an integrator circuit respond to an impulse function ? How does an integrator circuit respond to a ramp input [ tu (t)] ?. Answer: Convolution Integral. * Signifies convolution. h() signifies impulse response.

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Convolution

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  1. Convolution 3.1-3.2

  2. An Integrator Circuit

  3. Motivation • How does an integrator circuit respond to an impulse function? • How does an integrator circuit respond to a ramp input [tu(t)]?

  4. Answer: Convolution Integral * Signifies convolution h() signifies impulse response

  5. Impulse Response h(t) signifies the impulse response due to a unit impulse function, δ(t)

  6. Review of a Time Invariant System • Time invariant. A system is time invariant if a time shift of the input results in the same shift of the output

  7. Example 1Impulse Response of an TI System

  8. Example 2 =δ(t)

  9. Review of a Linear System • A system is linear if the principle of superposition applies

  10. An LTI System

  11. Example

  12. Generization (1) • If the weights of the input pulses in the input function vary as a function of time

  13. Generalization (2) (Δ→0) (Δ→0)

  14. Response of a Unit Step Function =u(t)

  15. Comparison

  16. Symmetrical Property

  17. Graphical Interpretation http://www.jhu.edu/~signals/convolve/ t=negative value!

  18. Graphical Interpretation t=positive value!

  19. Graphical Interpretation (Ymax) t=positive value!

  20. Graphical Examples • Impulse Response of an Integrator circuit due to the unit impulse function • Impulse Response of an integrator due to the unit step function • Impulse Response of an Integrator circuit due to the unit ramp input • Impulse Response of an integrator due to a rectangular pulse

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