1 HW 3: Solutions 1. The output of a particular system S is the time derivative of its input. a) Prove that system S is linear time-invariant (LTI). Solution:

2 HW 3: Solutions

3 b) What is the unit impulse response of this system? Solution: t  (t) 1/   t d/dt (  (t)) 1/  2  -1/  2 Limit as  tends to 0   unit impulse unit impulse response

4 HW 3: Solutions 2. Prove Property 5. That is, prove that, for an arbitrary LTI system, for a given input waveform x(t), the time derivative of its output is identical to the output of that system when subjected to the time derivative of its input. In other words, differentiation on the input and output sides are equivalent. Solution: Follows from Problem 1, and commutativity of convolution. Arbitrary LTI system d/dt x(t) y(t)y’(t) d/dt x(t) x’(t)y’(t)