Transfer Function from input to node.

Question

From the IIR filter flow graph below i don't understand how the transfer function is calculated in every node:

enter image description here

The circles contains 'X' inside are multiplications.
The circles contains 'Σ' inside are additions.
The triangles are delayers.

For example for node1, my textbook states that the transfer function is:

enter image description here

For node2:

enter image description here

For node3:

enter image description here

For node4:

enter image description here

I really don't understand how we have calculate this.

I don't think we'll understand it either unless you explain to us what the symbols mean. Are the crosses multiplications and the sigmas additions? And the triangles are diodes I presume? — Raskolnikov, May 29 '13 at 19:00
I don't know what tag fits this question, but [filters] is not about this sort of filters. If you had read the tag excerpt you'd know. — Asaf Karagila, May 29 '13 at 19:24
Minimus Hemimus, it is just powerpoint slides of the course. — 20317, May 29 '13 at 19:26
I suspect there is a mistake in your answer for node 1. There should be a minus in front of the power 2 of the last $z$ in the denominator. — Raskolnikov, May 29 '13 at 19:30

Raskolnikov · Accepted Answer · 2013-05-29T19:36:59.060

OK, I've had a try. You need to write down equations for what happens in the circuit. There is some feedback loop in the circuit which will help you solve the circuit in the end.

Let's call the state at the nodes $n_i$. Then, the state at $n_4$ is clearly the sum of $bx$ and what comes in from node 3 times some delay $z^{-1}$, so

$$n_4=z^{-1} n_3 + bx$$

Similarly, at node 2 we have

$$n_2 = bc_1 x -a_1 n_4$$

and at node 1 we have

$$n_1 = bc_2 x - a_2 n_4 \; .$$

Finally node 3 gives

$$n_3 = n_2 + z^{-1} n_1$$

This gives 4 linear equations in 4 unknowns. Solving them, I got the following for $n_1$

$$n_1=bx\frac{(c_2-a_2)+(c_2a_1-c_1a_2)z^{-1}}{1+a_1z^{-1}+a_2z^{-2}} \; .$$

Transfer Function from input to node.

1 Answers1