# Solving the transfer function

#### cfy30

Hi all,

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

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

Anyone has any idea? (Thinking)

Thanks,
cfy30

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#### Ackbeet

MHF Hall of Honor
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?

#### cfy30

I am not sure I get your idea. I still can't figure Y(s)/X(s) after differentiation.

(Crying)

cfy30

#### Ackbeet

MHF Hall of Honor
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:

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

Perhaps you could work with pulling the x(t) inside the integral, and doing a change of variable.

#### cfy30

The figure attached shows the system I am trying to model with the equation.

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

#### Ackbeet

MHF Hall of Honor
Are you sure you've written down your integral equation correctly? Where's the z(t) in it?

#### cfy30

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

#### CaptainBlack

MHF Hall of Fame
Your first system is not linear, so does not have a transfer function.

CB

Ackbeet

#### cfy30

Why the system is not linear?

I think the system is linear but time variant.

cfy30

#### cfy30

Let me take time variant back. I believe the system is also time invariant...