# Equivalent Diagram- How does counting paths let us eyeball?

• LongApple
In summary: The last figure shows the same path with a gain of x1.In summary, the youtube links are time stamped and the output is the sum of the inputs for the different circuits.
LongApple

All the youtube links are time stamped
1. Homework Statement

## 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 teh 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

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.

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.

Last edited:
NascentOxygen said:
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?

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.

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

LongApple said:
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

LongApple said:
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.

## 1. What is an equivalent diagram?

An equivalent diagram is a visual representation of a mathematical problem or concept. It uses symbols, shapes, and lines to help visualize and understand the problem or concept.

## 2. What is counting paths in an equivalent diagram?

Counting paths in an equivalent diagram refers to the process of determining the number of possible ways to move from one point to another in the diagram. This can help in solving problems involving permutations, combinations, and probability.

## 3. How does counting paths help us "eyeball" in an equivalent diagram?

Counting paths allows us to quickly estimate the number of possible solutions or outcomes in a problem by looking at the diagram. This is known as "eyeballing" because we can make a quick, visual estimation without needing to do complex calculations.

## 4. What types of problems can be solved using counting paths in an equivalent diagram?

Counting paths can be applied to a variety of problems, including those involving permutations, combinations, and probability. It is particularly useful in problems with multiple variables or options, as it allows for a visual representation of all the possible outcomes.

## 5. Are there any limitations to using counting paths in an equivalent diagram?

While counting paths can be a helpful tool in solving mathematical problems, it is not always a precise method and may not provide the exact solution. It is important to use other methods and techniques to verify the accuracy of the solution obtained through counting paths.

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