Lattice structure realization of an all-pole system

Question

Given an IIR system with the system function,

$$ H(z) = \frac{1}{2 + 1.4z^{-1} + 1.8z^{-2}} $$

I need to draw the lattice structure realization.

I have drawn the lattice structure for $\frac{1}{1 + 0.7z^{-1} + 0.9z^{-2}}$, which is shown in the figure below.

Now since $H(z) = \frac{1}{2(1 + 0.7z^{-1} + 0.9z^{-2})} $, should I multiply the signal $x(n)$ by $2$ or $1/2$ before feeding to the system? My intuition is that I should multiply with $\frac 1 2$, but then since $2$ appears as the multiple in the denominator polynomial, I got confused.

Answer 1

As mentioned by LamebrainEddy, you should find the difference equation of your system. To do so, recall that:

$H(z) = \frac{Y(z)}{X(z)} = \frac{1}{2 + 1.4z^{-1} + 1.8z^{-2}}$

So:

$Y(z)(2 + 1.4z^{-1} + 1.8z^{-2}) = X(z)$

$2Y(z) + 1.4z^{-1}Y(z) + 1.8z^{-2}Y(z) = X(z)$

And if you anti-transform you get:

$2 y[n]+1.4 y[n-1]+1.8 y[n-2]$ = x[n]

Finally, solving for $y[n]$:

$y[n]=-0.7y[n-1]-0.9y[n-2]+0.5x[n]$

From this equation you can notice that x[n] is scaled by 0.5

Answer 2

Rearrange into a difference equation form and your answer will become clearer!