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A system is known to be LTI. The response of the system to a step function u[n] is δ[n] + δ[n-1].
a.) Find the response of the system to 2u[n] + u[n-1]
b.) Find the response to the unit impulse δ[n].
My professor hasn't been doing a good job of explaining this concept so far, please help!
a.) Find the response of the system to 2u[n] + u[n-1]
If you have a linear system described by a response H, then you know that
\$ H(\alpha{}x_1 + \beta{}x_2) = \alpha{}H(x_1) + \beta{}H(x_2)\$.
Since the input you're being asked about is a linear combination of unit step functions, and you know the response to a unit step function, you can work out the response from this principle.
b.) Find the response to the unit impulse δ[n].
The Kronecker delta function used in discrete time systems has value 1 at n=0 and 0 for all other n. Therefore you can write
\$ \delta[n] = u[n] - u[n-1] \$,
and then you can use linearity again to find the system response.
