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In particular I am having trouble with 6b).

From what I understand, we can split a difference LTI equation into two sums, the sum of the previous responses, and the sum of the previous inputs. Something like this:

Now to get a zero-state response, I have to zero out the first part of the formula. This I understand.

So to get the zero-state step response, I must use x[n] = u[n] and zero out all past outputs of the system.

The problem I'm having is deriving y[n] in the first place.

I have had a couple ideas, however am not confident in their correctness:

  • Since H(z) = Y(z)/X(z) I could compare the given H(z) to Y(z)(1 - 1/z) because that's the z-transform of 1/u[n]. Correct me if I'm wrong. Then I could make Y subject of the formula and figure out the inverse transform from there.

  • I thought about using convolution with the inverse transform of
    H(z) and input signal u[n].

However I'm not convinced these are correct because nowhere am I "enforcing" the definition of zero-ing out the previous outputs.

Any guidelines would be appreciated!

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HINT:

The (zero-state) step response is just the cumulative sum of the impulse response $h[n]$:

$$y_{ZS}[n]=u[n]\sum_{k=0}^nh[k]$$

This follows in a straightforward manner from the convolution of a unit step $u[n]$ with the impulse response:

$$y_{ZS}[n]=(h\star u)[n]$$

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  • $\begingroup$ Would you mind elaborating further on how to obtain the first statement? Particularly I don't understand how this relates to being the "zero state". I understand the use of the convolution sum, but not how it relates to the zero state. Thank you :) $\endgroup$ – Novicegrammer Nov 5 at 18:46
  • $\begingroup$ @Novicegrammer: By definition, the impulse response $h[n]$ is the "zero-state" response to an impulse. So any "zero-state" response is obtained by convolving the respective input signal with the impulse response. $\endgroup$ – Matt L. Nov 5 at 21:47

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