ECE Solution 301: Quiz 4 Purdue University, Summer 2009 1. Recall that F {x(t)} = x(t)ejt dt. Compute the Fourier transform F {et u(t)}, where > 0 Solution:
F {et u(t)} =
et u(t)ejt dt et(+j) dt
0
= =
1 + j
2. Let an LTI system have impulse response h(t) = e2t u(t). Without using convolution, use your answer from the previous problem to nd output the y (t) when x(t) = e5t u(t) is input. Solution: Work in the frequency domain, then take inverse Fourier transform: y (t) = x(t) h(t) Y ( ) = X ( )H ( ) = F {e5t u(t)}F {e2t u(t)} 1 1 = (from problem 1). 5 + j 2 + j Using partial fraction expansion, we nd 1/3 1/3 + 5 + j 2 + j 1 2t 1 y (t) = e u(t) e5t u(t) (from problem 1). 3 3 Y ( ) =
1