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I have this system and tried to write down its difference equation as follows

$$ y[n] = b_0v[n] + b_1v[n-1]\\ v[n] = x[n] - a_1v[n-1] $$

I was asked to prove that the system can be described by the following difference equation

$$ y[n] + a_1y[n − 1] = b_0x[n] + b_1x[n − 1]$$

Please help me how to prove it.

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HINT:

Just express $y[n]$ and $x[n]$ in terms of $v[n-k]$, $k=0,1,\ldots$, and show the equality.

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  • $\begingroup$ I'm sorry, i still don't understand how, can you please explain $\endgroup$ – Adam Apr 16 '19 at 12:21
  • $\begingroup$ @Adam: Well, you have two equations involving $y[n]$, $x[n]$, and $v[n-k]$, so you can express $y[n]$ and $x[n]$ in terms of $v[n-k]$ and then show the given equality. $\endgroup$ – Matt L. Apr 16 '19 at 12:28

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