Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal...
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Transcript of Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal...
Convolution ExamplesEx2.9:Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)
Calculation of Impulse Response IllustrationThe governing equation of the RC circuit below is of the form:
Calculation of Impulse Response Illustration
The charge stored in C due to this initial current flow is::
Calculation of Impulse Response Illustration
Convolution in terms of Step Response
• Step response is simply the integral of the impulse response Employ the step response of a system to express its response to an arbitrary input signal.
Use Integration by parts:
With u=x(t-ג) dv = h(ג) d ג therefore:
Is simply the step response a(t) and
Convolution in terms of Step Response
Substituting these expressions and integrating by parts we get:
The system is initially unexcited, so that a(-∞)=0 and x(t-ג)| 0==∞ג
Thus the first term is zero and final result is: