1. Convolution Examples Ex2.9:
Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

2. Calculation of Impulse Response Illustration
The governing equation of the RC circuit below is of the form:

3. Calculation of Impulse Response Illustration
The charge stored in C due to this initial current flow is::

4. Calculation of Impulse Response Illustration

5. 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

6. 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)| ג=∞=0
Thus the first term is zero and final result is: