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Signal Processing

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  • $\begingroup$ i'm gonna have to rewrite your whole question in the language that electrical engineers doing DSP use. then answer it. we say "$x[n]$" instead of "$x_t$" and you'll have to be more explicit about what you mean by "$\Delta y_t$, but i think you mean $y[n]-y[n-1]$. the initial conditions are what we might call "$x[-1]$" and "$y[-1]$". and, i think you might have to define $G(\cdot)$. $\endgroup$ Commented Nov 19, 2020 at 2:35
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    $\begingroup$ i have to say, even with the bounty, i dunno that this is worth my time detangling. there is so much to sort out just expressing the problem in clear, well-defined mathematical terms with notational convention consistent with electrical engineering DSP. $\endgroup$ Commented Nov 19, 2020 at 4:20
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    $\begingroup$ If the system is stable (i.e. not marginally stable or unstable), the final value should not depend on initial conditions. Am I wrong with this assumption? $\endgroup$ Commented Nov 20, 2020 at 15:57
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    $\begingroup$ This particular system's Y will not go to zero even if X eventually goes to zero. But it is stable (with appropriate psi and phi) in the sense that it settles to a finite value so long as X remains constant. The final value depends on the initial conditions. $\endgroup$ Commented Nov 20, 2020 at 19:06
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    $\begingroup$ Your expression for $H(z)$ is not consistent with your description of the problem. In particular, $H(z)$ contains both $\psi$ and $\phi$, yet you do not mention $\phi$ in your problem statement -- it just crops up at the very end. Moreover, I went to the effort to try a couple of guesses about what $F_\phi$ and $G_\psi$ really were, and neither ended up with the same $H(z)$. Stackexchange wants us to give answers to the question that was actually asked, and I can't tell what that is at the moment. Try clarifying your question, preferably by simplifying it. $\endgroup$ Commented Nov 21, 2020 at 17:01