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: