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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 3.1-3.2
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
Impulse Response h(t) signifies the impulse response due to a unit impulse function, δ(t)
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
Example 2 =δ(t)
Review of a Linear System • A system is linear if the principle of superposition applies
Generization (1) • If the weights of the input pulses in the input function vary as a function of time
Generalization (2) (Δ→0) (Δ→0)
Graphical Interpretation http://www.jhu.edu/~signals/convolve/ t=negative value!
Graphical Interpretation t=positive value!
Graphical Interpretation (Ymax) t=positive value!
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