e1sample.pdf

# e1sample.pdf - 332:345 – Linear Systems Signals – Fall...

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332:345 – Linear Systems & Signals – Fall 2009 Sample Exam-1 Questions 1. The impulse response h(t) of an LTI system and an input signal f(t) are nonzero over the following time ranges: h(t), a t b f(t), c t d The corresponding output is given by the convolutional equation: y(t) = h(t τ)f(τ)dτ Determine the time range for y(t) , and the precise limits of the above integral. 2. Sketch the two signals h(t) = e t u(t) and f(t) = u(t) u(t 5 ) . Then, determine their convolution y(t) = h(t τ)f(τ)dτ using the following two methods: (a) By performing the indicated time integration. (b) By working with Laplace transforms. 3. Using Laplace transforms, solve the following differential equation, ¨ y(t) + 5 ˙ y(t) + 4 y(t) = 3 f(t) where f(t) = e 3 t u(t) with arbitrary initial conditions: y( 0 ) and ˙ y( 0 ) . Identify the parts of the solution that correspond to the decomposition of y(t) into a “homogeneous solution" and a “particular solution”. Similarly,
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