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

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

is given. They integrate each side twice which yields

$$a_2y(t)+a_1y_{(-1)}(t)+a_0y_{(-2)}(t)=b_2x(t)+b_1x_{(-1)}(t)+b_0x_{(-2)}(t)$$

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

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.

http://www.photo-host.org/img/719732diagram.gif

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

Edit: corrected LaTeX code.

 

[mighty2000]

Hello there! As a scientist who has worked with block diagrams extensively, I understand how they can be confusing at first. Let me try to explain how to read the "Direct Form II" block diagram in the context of the given second-order differential equation.

First, let's understand the overall structure of the block diagram. The two integrators on the left represent the two integrals of the second derivative of y(t), while the two integrators on the right represent the two integrals of the second derivative of x(t). The arrows indicate the flow of information, with the input x(t) feeding into the first integrator on the right and the output y(t) being the final result on the left.

Now, let's focus on the individual blocks and their functions. The integrators essentially perform the integration operation on the input signal, with the output being the integral of the input. The summing junctions, represented by the circles with a plus sign, add the signals coming into them. The coefficients a0, a1, a2, b0, b1, and b2 are multiplied with the respective signals before being added at the summing junctions.

To read the diagram, start from the input x(t) and follow the arrows. The signal passes through the first integrator, then goes through the summing junction where it is multiplied by b0 and added to the integral of the second derivative of y(t). This sum is then multiplied by b1 and added to the integral of the first derivative of y(t). Finally, this sum is multiplied by b2 and added to the integral of y(t), giving us the output y(t).

I hope this explanation helps in understanding the "Direct Form II" block diagram. Keep in mind that there may be slight variations in notation and structure for different realizations, but the basic principles remain the same. Let me know if you have any further questions and I'll be happy to assist. Best of luck with your studies!