LTI Response Help

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!

1 Answer

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.

• Hi, so is a. simply 2δ[n]+2δ[n-1]+δ[n-1]+δ[n-2] ??? – Shankar Kumar Mar 11 '13 at 4:52
• You can do one more simplification step, but that's basically it. – The Photon Mar 11 '13 at 4:58
• Combining like terms --> 2δ[n]+3δ[n-1]+δ[n-2] – Shankar Kumar Mar 11 '13 at 4:59
• You got it now. – The Photon Mar 11 '13 at 5:07