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Equivalent Diagram- How does counting paths let us eyeball?

  1. Jan 28, 2015 #1


    All the youtube links are time stamped



    1. The problem statement, all variables and given/known data



    upload_2015-1-28_2-22-23.png


    2. Relevant equations


    3. 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
     
  2. jcsd
  3. Jan 28, 2015 #2

    NascentOxygen

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    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.
     
  4. Jan 28, 2015 #3

    rude man

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    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: Jan 29, 2015
  5. Jan 28, 2015 #4
    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?
     
  6. Jan 29, 2015 #5

    NascentOxygen

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    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.
     
  7. Jan 29, 2015 #6
    So the reason we know it is equivalent by eyeballing is that they all sum to 4X1; 4X2?
     
  8. Jan 30, 2015 #7

    NascentOxygen

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    Don't forget X. The outputs are all X + 4X1 + 4X2
     
  9. Feb 4, 2015 #8

    NascentOxygen

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    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.

    upload_1a.jpg
     
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