I'm having trouble on how to check if a given difference equation corresponds to an LTI system. For instance,

Does the following difference equation correspond to a LTI DT system ? Explain. $$y[n+3]+3y[n]+y[n+1]=\left ( \frac 12 \right )^{n+3}u[n+3]$$

I've learned how to show a system is LTI given an explicit statement of the input and the output of a system. However in this case, I am not given any relationships. Are there any assumptions I should be making about LTI systems such as causality? Any help will be much appreciated.

  • $\begingroup$ I'm assuming $u[n]$ is the usual step function. You can have causal and non-causal LTI systems, that detail is not important here. Do you know intuitively how to test for time invariance? $\endgroup$ – Engineer Feb 17 '20 at 18:45
  • $\begingroup$ The general way x(t-to) -> y(t-to). $\endgroup$ – Nick Yarn Feb 17 '20 at 18:47
  • $\begingroup$ $x[n-n_o] \rightarrow y[n-n_o]$ $\endgroup$ – Nick Yarn Feb 17 '20 at 18:54
  • $\begingroup$ Without equations, it is asking the question: if a give an input and get an output, does the shifted version of the input result in the same output except shifted? If that is true, then the system is time invariant. You already wrote down the equation for the output, now everywhere you see $x[n]$ in the equation, replace it with $x[n-n_0]$, and check is it equal to the shifted version of the output? $\endgroup$ – Engineer Feb 17 '20 at 18:55
  • 1
    $\begingroup$ But there is no x[n] in the equation. $\endgroup$ – Nick Yarn Feb 17 '20 at 19:00

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