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Block Diagram

I arrived at:

$$\frac{Y[n]}{X[n]}=1-\frac{1}{6}z^{-2}+\frac{5}{6}z^{-1}$$

How do I arrive at:

$$H(z)=\frac{1}{1-\frac{5}{6}z^{-1}+\frac{1}{6}z^{-2}}$$

ocrdu
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uncutappleslices
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  • Looks like [this](https://i.stack.imgur.com/Q62zq.png) in Python. Did you write up the equation to solve, correctly? – jonk Oct 06 '22 at 09:27
  • @jonk The equation I got may be wrong but the H(z) is correct – uncutappleslices Oct 06 '22 at 09:32
  • Yes. I agree. And the Python code I wrote also agrees with that. – jonk Oct 06 '22 at 09:53
  • @jonk how do you form the equation from the block diagram mathematically? – uncutappleslices Oct 06 '22 at 10:08
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    You can see how I did. It's right there in the picture if you click on it. Just read it. I said, *"Yn is equal to 5/6 times Yn divided by z plus Yn divided by z twice over and then times (-1/6) added to Xn."* That's all I wrote there. Then I asked Python to do the darned algebra for me, which I was too lazy to bother with. – jonk Oct 06 '22 at 10:12
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    Vivek, feel free to work through the algebra by hand. I generally don't do much of that anymore. There was a time back when we had slide rules and mathematical tables of logarithms to work from because a calculator cost USD1000 when that meant something (like the same as buying a car.) But times have changed and we have SymPy and Python tools and while it is still important to know how to do the algebra by hand should you be stranded on a desert island somewhere, there's no reason to be forced to use paper and pencil and worry about typos, today. – jonk Oct 06 '22 at 10:21

2 Answers2

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The first equation is not right. It should be

$$Y[n]=X[n]-\frac{1}{6}z^{-2}Y[n]+\frac{5}{6}z^{-1}Y[n]$$

Then if follows that

$$Y[n]-\frac{5}{6}z^{-1}Y[n]+\frac{1}{6}z^{-2}Y[n]=X[n]$$

$$Y[n](1-\frac{5}{6}z^{-1}+\frac{1}{6}z^{-2})=X[n]$$

$$\frac{Y[n]}{X[n]}=\frac{1}{1-\frac{5}{6}z^{-1}+\frac{1}{6}z^{-2}}$$

Suba Thomas
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2

Simply write one ADDER output at one time, simple "blocks" functions.
Each delay block output is the "input" * (z^-1).
The output of the right delay block is Yn* (z^-1).
The output of the left delay block is Yn* (z^-2).

Output of 1st adder \$ S1 = Xn -1/6 * z^-2 * Yn \$.
Output of 2nd adder \$ Yn= S1 + 5/6 * z^-1 * Yn \$.

Replacing S1,
So, \$ Yn = Xn -1/6 * z^-2 * Yn + 5/6 * z^-1 * Yn \$.

Collecting Yn : \$ Yn * (1+ 1/6 * z^-2 - 5/6 * z^-1) = Xn \$ ...

Then \$ Yn/Xn = H(z) = 1/ ( 1+ 1/6 * z^-2 - 5/6 * z^-1) \$.

Antonio51
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