This preview has intentionally blurred sections. Sign up to view the full version.View Full Document
Unformatted text preview: ECE 314 Solution to Test #1 Georghiades Given: September 26, 2008. Instructions : Exam is open-book, open notes. Please give your solution in the allocated space. If more space is needed, use the back of the page. All problems carry equal weight. 1. Consider a linear, time-invariant system. It is known that when the input is a unit-step, the output is given by 1 2 (1- e- 2 t ) u ( t ). Find the output, y ( t ), when the input is x ( t ) = e- t u ( t ) and compute its energy. Solution The impulse response, h ( t ), of the system is the derivative of the unit-step response: h ( t ) = d dt bracketleftbigg 1 2 (1- e- 2 t ) u ( t ) bracketrightbigg = 1 2 (1- e- 2 t ) δ ( t ) + e- 2 t u ( t ) = e- 2 t u ( t ) . Then the out of the system when the input is x ( t ) is y ( t ) = x ( t ) * h ( t ), i.e., y ( t ) = integraldisplay ∞-∞ e- 2 τ u ( τ ) e- ( t- τ ) u ( t- τ ) dτ = e- t integraldisplay ∞-∞ e- τ u ( τ ) u ( t- τ ) dτ = e- t u ( t ) integraldisplay t e- τ dτ = parenleftBig...
View Full Document
- Spring '08
- LTI system theory, Impulse response