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Math Help - Solving the transfer function

  1. #1
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    Solving the transfer function

    Hi all,

    I have been trying to solve the follow question but with no success. Anyone has idea on finding H(s)?

    Solving the transfer function-transferfunction_1.jpg

    In one special case, the equation can be written as follow but again I have no clue on solving it.

    Solving the transfer function-transferfunction_3.jpg

    Anyone has any idea?


    Thanks,
    cfy30
    Attached Thumbnails Attached Thumbnails Solving the transfer function-transferfunction.jpg   Solving the transfer function-transferfunction_2.jpg  
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  2. #2
    A Plied Mathematician
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    If you know that x(t) is differentiable, then I would differentiate the whole thing, take the Laplace Transform, and solve for Y/X. Is anything preventing you from doing that?
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  3. #3
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    I am not sure I get your idea. I still can't figure Y(s)/X(s) after differentiation.

    Solving the transfer function-diff.jpg




    cfy30
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  4. #4
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    Yeah, I see what you mean. This is a doozy of a problem. Is this a textbook problem? If so, there's probably some trick you're supposed to see in order to solve it. I tried doing the Laplace Transform directly, and then fiddling around with interchanging the order of integration. You can get some interesting equations that way, but not towards getting the final ratio desired. And you can try tricks with integration by parts, but that ends up doing the same thing as interchanging the order of integration. One thought that did occur to me was this: convolution. If you were to focus only on the LT of the integral term with the x(t) multiplying it, I wonder if you couldn't use the convolution theorem to help you out there. It'd be worth trying, because then you might get something like this:

    Y(s)=X(s)-A\,X(s)Y(s)\times\,\text{something}.

    Perhaps you could work with pulling the x(t) inside the integral, and doing a change of variable.
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  5. #5
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    The figure attached shows the system I am trying to model with the equation.

    Solving the transfer function-system.jpg

    It is a feedback system to suppress x(t) at the node y(t). The system itself is quite simple but the transfer function is giving me headache. What I want to do is formulating H(s) and then determine the optimum A if the optimum A exists......


    cfy30
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  6. #6
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    Are you sure you've written down your integral equation correctly? Where's the z(t) in it?
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  7. #7
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    I am sure the equation I have is correct. z(t) is the "wanted" signal and x(t) is the interference that needs to be canceled. When the system starts to run, x(t) will be suppressed, leaving z(t) as the output y(t). Imagine z(t) is cos(2*pi*200*t) and x(t) is cos(2*pi*50*t). y(t) contains cos(2*pi*200*t) only and is free of cos(2*pi*50*t).


    cfy30
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  8. #8
    Grand Panjandrum
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    Your first system is not linear, so does not have a transfer function.

    CB
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  9. #9
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    Why the system is not linear?

    I think the system is linear but time variant.


    cfy30
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  10. #10
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    Let me take time variant back. I believe the system is also time invariant...
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  11. #11
    Grand Panjandrum
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    Quote Originally Posted by cfy30 View Post
    Why the system is not linear?

    I think the system is linear but time variant.


    cfy30
    Consider two signals x_1(t) and x_2(t) and corresponding outputs y_1(t), y_2(t). Now what is the output when the input is z(t)= x_1(t)+x_2(t)? (note I am here working with a necessary condition for linearity not the full condition, to show that linearity fails it is sufficient to show that a necessary condition fails)

    CB
    Last edited by CaptainBlack; July 21st 2010 at 08:11 AM.
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  12. #12
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    Did you mean "corresponding outputs y_{1}(t) and y_{2}(t)."?
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  13. #13
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    My thinking is, let x(t) = x1(t) and z(t) = x2(t), x1(t) and x2(t) have different frequencies.
    Output of the system, y(t) is always equal to z(t) or x2(t). The system behaves as a notch filter. That is what I say the system is LTI.


    cfy30
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  14. #14
    Grand Panjandrum
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    Quote Originally Posted by Ackbeet View Post
    Did you mean "corresponding outputs y_{1}(t) and y_{2}(t)."?
    Yes, its my fault for not reading the spell-checker suggestions carefully before accepting
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  15. #15
    Grand Panjandrum
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    Quote Originally Posted by cfy30 View Post
    My thinking is, let x(t) = x1(t) and z(t) = x2(t), x1(t) and x2(t) have different frequencies.
    Output of the system, y(t) is always equal to z(t) or x2(t). The system behaves as a notch filter. That is what I say the system is LTI.


    cfy30
    An LTI system has to be LINEAR as well as time invariant, see the Wikipedia article. What you have written above is gobbledygook.

    CB
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