How Can I Linearize \( f(t) = \sin(\Phi(t)) \) Using Laplace Transform?

[Jarfi]

By using the laplace transform:

$f(t)=sin(Φ(t))$

I want it in the form:

F(S)/Φ(S)

The purpose is to linearize it in order to put it into a larger transfer function, so far my only solution is to simplify it using taylor expansion.

 

[jasonRF]

Yes, you can linearize - just Taylor expand about the relavent point. However, since you really care about the Laplace transform of this, the linear approximation will only help you if $\phi(t)$ changes very little for $0<t<\infty$.

 

[Jarfi]

Yes, you can linearize - just Taylor expand about the relavent point. However, since you really care about the Laplace transform of this, the linear approximation will only help you if $\phi(t)$ changes very little for $0<t<\infty$.

Taylor expansion the sin function would yield a polynomial of several orders, say I had two 3 order approximated sinus functions in series(with an output between them), then I have a 6th order polynomial, increasing complexity. So i was hoping for another solution.

Φ is from 0-45°

 

[jasonRF]

I don't understand what you mean. A linear approximation is, by definition, linear. How would it be 6th order?

If you want a linear approximation, it obviously must be of the form $\sin\phi(t) \approx a \, \phi(t) + b$. How you select $a$ and $b$ depends upon what criterion makes the most sense for the problem you are trying to solve. Unless you tell us what problem you are actually trying to solve I doubt I can help much more.

jason

 

[Jarfi]

I don't understand what you mean. A linear approximation is, by definition, linear. How would it be 6th order?

If you want a linear approximation, it obviously must be of the form $\sin\phi(t) \approx a \, \phi(t) + b$. How you select $a$ and $b$ depends upon what criterion makes the most sense for the problem you are trying to solve. Unless you tell us what problem you are actually trying to solve I doubt I can help much more.

jason

It's linear in the frequency domain, NOT in the time domain: My preferrence is exactly not to approximate it, but to convert it. I already know how to approximate-linearize a sinus function

 

[jasonRF]

It's linear in the frequency domain, NOT in the time domain: My preferrence is exactly not to approximate it, but to convert it. I already know how to approximate-linearize a sinus function

By using the laplace transform:

$f(t)=sin(Φ(t))$

I want it in the form:

F(S)/Φ(S)

The purpose is to linearize it in order to put it into a larger transfer function, so far my only solution is to simplify it using taylor expansion.

I still don't know what you are asking - you use the word "it" a lot, and now it is clear that the word "it" may refer to different things in different places. So … what exactly are you trying to approximate and in what form? What do you mean by "convert it"? When you say linear, do you mean a linear function of $\Phi(s)$, or a linear function of $s$?

I would be happy to help if you answer these questions, or (probably better) just state a more explicit question that helps us understand what you are doing.

jason