Equivalent Diagram- How does counting paths let us eyeball?

[LongApple]

All the youtube links are time stamped
1. Homework Statement

Homework Equations

The Attempt at a Solution

I wrote it out the operator expressions each by hand and got the same result but I don't understand how he is able to just eyeball it. I'm trying to develop some intuition.

a. To start, why is he counting distinct signal paths paths? What is the motivation to see why this information may be useful in letting us eyeball which block diagrams are the same. He mentioned earlier in the video that this could be useful but I didn't understand.

b.

He says something about "making a sum" and seeing "how many of them have the same sum" What does that mean? He then looks for the paths with the biggest delay. I don't see why this info would help us other than to maybe to disprove two block diagrams are different.
^ See time stamped youtube vid

At the very end, he is able to conclude that the answer is 3 but I don't see how we have proof

 

[NascentOxygen]

In (a) there are 4 paths: X; 4X₂; 2X₁; another 2X₁
In (b) there are 3 paths: X; 4X₂; 4X₁
In (c) there are 3 paths: X; 4X₁; 4X₂

The output Y is the sum of these. The result: all are equivalent.

I have used X₁ to denote X after one delay

BTW, I haven't looked at the videos.

 

[rude man]

Call the output of every summing junction z
Then,for the 1st circuit,
z[n] = x[n] +2x[n-1]
and y[n] = z[n] + 2z[n-1]
then y[n] = x[n] + 2x[n-1] + 2x[n-1] + 4x[n-2]
= x[n] + 4x[n-1] + 4x[n-2]
You can proceed likewise for the 2nd and 3rd diagram to show y[n] is the same for all three.

 

[LongApple]

In (a) there are 4 paths: X; 4X₂; 2X₁; another 2X₁
In (b) there are 3 paths: X; 4X₂; 4X₁
In (c) there are 3 paths: X; 4X₁; 4X₂

The output Y is the sum of these. The result: all are equivalent.

I have used X₁ to denote X after one delay

BTW, I haven't looked at the videos.

I have used X₁ to denote X after one delay

So for example then, what does 4X₂; 4X₁ mean? Aren't there 9 paths because you have 4, 4, and 1? What is 4X2 for example? So it seems like based on your notation that would mean 4 times X after 2 delays in part a). Where does this 4 times X after two delays come from?

Is this his method of eyebaling?

 

[NascentOxygen]

Those triangles with a number inside denote an amplifier (e.g., a voltage amplifier), they have no effect on the delay.
So I used 4X₂ to denote X that has passed through two delays and has had its amplitude multiplied by 4. The order in which that has happened is irrelevant.

The circle with a cross in it represents a summer, its output is the sum of the inputs.

 

[LongApple]

So the reason we know it is equivalent by eyeballing is that they all sum to 4X1; 4X2?

 

[NascentOxygen]

So the reason we know it is equivalent by eyeballing is that they all sum to 4X1; 4X2?

Don't forget X. The outputs are all X + 4X1 + 4X2

 

[NascentOxygen]

To start, why is he counting distinct signal paths paths?

Finally I'm at my desktop so can edit your first image to highlight the 4 different paths. Each path delivers X (after some transformation) to the output.

The first shows X after two delays and two gains of x2. The middle figures show two different paths delivering X with a delay and a gain of x2. The lower figure shows a straight-through path delivering X at Y.