Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal...

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

Transcript of Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal...

Page 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)

Convolution ExamplesEx2.9:Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

Page 2: Convolution Examples Ex2.9: 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:

Page 3: Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

Calculation of Impulse Response Illustration

The charge stored in C due to this initial current flow is::

Page 4: Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

Calculation of Impulse Response Illustration

Page 5: Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

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

Page 6: Convolution Examples Ex2.9: Consider the convolution of a ramp and the decaying exponential signal of the form x(t) =exp(-αt)u(t)

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: