HW #3

Due Tuesday, January 26

1. For a Linear, Time-Invariant system, we know that the output y(t) is the convolution of the impulse response h(t) with the input w(t). That is, ¬ ugly convolution

a) Use the two-sided Laplace Transform to show that:

¬ beautiful multiplication

b) Find the impulse response for the following circuit:

c) Use Laplace transforms to find H(s) then find y(t) due to the following inputs:

i)

ii)

iii)

d) Find the refion of convergence of X(s) and state whether or not we can use the final value theorem to fine x(¥ ).

i) ii) iii)

iv) v) vi)

e) For X(s) in problem 1d), state whether or not the Fourier transform exists and if we can find X(f) by substituting s=j2p f into X(s). If we can use this substitution, then find X(f).

2.a) Find the Laplace transform of the following signals which are periodic after t=0:

b) Find x(t) and then sketch if

i) ii) iii)

c) Use the two-sided Laplace transform to find X(s) for the following signals (be sure to specify the ROC):

i) ii)