Trouble understanding block diagrams

cyph1e

Hello everyone!

I'm currently reading Signals, Systems and Transforms by Charles L Phillips, John Parr and Eve Riskin, and I can't really get a grip of how to read a specific block diagram.

The second-order differential equation

[tex]a_2 \frac{d^2y(t)}{dt^2}+a_1 \frac{dy(t)}{dt}+a_0 y(t)=b_2 \frac{d^2x(t)}{dt^2}+b_1 \frac{dx(t)}{dt}+b_0 x(t)[/tex]

is given. They integrate each side twice which yields

[tex]a_2y(t)+a_1y_{(-1)}(t)+a_0y_{(-2)}(t)=b_2x(t)+b_1x_{(-1)}(t)+b_0x_{(-2)}(t)[/tex]

where the notation [itex]y_{(k-n)}(t)[/itex] indicates the nth integral of the kth derivative of [itex]y(t)[/itex].

They present two types of block diagrams called "Direct Form I" and "Direct Form II", and I don't know if that is standard for these types of realizations. I understand "Direct Form I" perfectly well, but I'm really having some trouble understanding the second form as I can't really seem to follow the arrows, get a grip of where to start and derive an equation etc.

Here is the realization of the above equation using "Direct Form II", I'd be very glad if someone could walk me through how to read it.



(The bottom diagram is a simplification of the top diagram with two integrators eliminated.)

Edit: corrected LaTeX code.