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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!

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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.

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

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