This is the transfer function for the Direct Form 1 of the IIR Filter:
I'm trying to understand the rules to convert $H(z)$ into its corrispondent structure
I've just understood how to draw $H_1(z)$, but i don't understand why $H_2(z)$ is drawed as such in diagram. Can you explain to me ?
ps: can i also implement a FIR filter using Direct Forms 1 and 2 ?
You can't implement the transfer function H(z) directly, you need to convert it to a difference equation. However, the process is trivial, so once you understand it you'll see the connection between the diagram and transfer function better.
First, we need to unroll the summation. For example, we get this with m=2, for a second-order equation:
$H(z) = \frac{b_{0} z^{0} + b_{1} z^{-1} + b_{2} z^{-2}}{a_{0} z^{0} + a_{1} z^{-1} + a_{2} z^{-2}}$
As the transfer function is the relationship of the system's output to input, we start by making that substitution.
$H(z) = \frac{Y(z)}{X(z)}$
$\frac{Y(z)}{X(z)} = \frac{b_{0} z^{0} + b_{1} z^{-1} + b_{2} z^{-2}}{a_{0} z^{0} + a_{1} z^{-1} + a_{2} z^{-2}}$
The output is Y, input is X, so this shows that for the fraction on the right, the top part is related to the input and bottom is related to output. Let's rearrange to get output in terms of input.
$Y(z)(a_{0} z^{0} + a_{1} z^{-1} + a_{2} z^{-2}) = X(z)(b_{0} z^{0} + b_{1} z^{-1} + b_{2} z^{-2})$
Here we multiply the terms and rearrange to get our difference equation in terms of y[n], the output now.
$a_{0} y[n] + a_{1} y[n-1] + a_{2} y[n-2] = b_{0} x[n] + b_{1} x[n-1] + b_{2} x[n-2]$
$a_{0} y[n] = b_{0} x[n] + b_{1} x[n-1] + b_{2} x[n-2] - a_{1} y[n-1] - a_{2} y[n-2]$
$y[n] = \frac{b_{0}}{a_{0}} x[n] + \frac{b_{1}}{a_{0}} x[n-1] + \frac{b_{2}}{a_{0}} x[n-2] - \frac{a_{1}}{a_{0}} y[n-1] - \frac{a_{2}}{a_{0}} y[n-2]$
And you can see why we typically "normalize" the coefficients by dividing all by a0 to get our final form. I use this prime notation to show that the coefficients are normalized ($b^{'}_{0}=\frac{b_{0}}{a_{0}}$) as compared to the above, but we typically assume normalized coefficients.
$y[n] = b^{'}_{0} x[n] + b^{'}_{1} x[n-1] + b^{'}_{2} x[n-2] - a^{'}_{1} y[n-1] - a^{'}_{2} y[n-2]$
So, in retrospect you can get some intuition of the block diagram by looking at the transfer function. The summations give the strings of delays and their coefficients, and the denominator becomes subtracted terms in the difference equation.
It would help to apply some specificity to the general equation:
$H(z) = \frac{b_0 + b_1z^{-1}+b_2z^{-2}}{1+a_1z^{-1}+a_2z^{-2}}$
Rearrange into a difference equation, skipping the steps for brevity:
$y[n] = b_0x[n] +b_1x[n-1]+b_2x[n-1]-a_1y[n-1]-a_2y[n-2]$
Does the difference equation form show now that the Direct Form structure is derived purely by inspection?
