How to Find the Laplace Transform of cos(t) * f(t)?

[carlodelmundo]

I am given a function f(t) with it's corresponding Laplace Transform in the Frequency Domain (F(s)).

I'm having a hard time wrapping my head around the product of say, L{cos(t)*f(t)}. The * is multiplication and not convolution. Must I do the integration for the Laplace transform by hand, or is there a short cut method using the table of Laplace transforms?

I want hints/direction, not a definitive answer.

Thanks

 

[MisterX]

The convolution theorem works both ways; the Laplace transform of a product is the convolution of the Laplace transforms of the multiplicands. This may not be particularly useful to you though.

For cos(t)f(t) in particular, you might try expressing the cosine in terms of exponential functions.

It may be worth noting that cos(t)f(t) is the "double-sideband suppressed-carrier" form of amplitude modulation.

 

[carlodelmundo]

Thanks for the response. After mulling it over, I've figured it out:

One must treat f(t) in cos(t) * f(t) as simply any function (it doesn't matter what it is). After converting the cos(t) to exponentials (through) identities, one realizes that cos(t) affects f(t) through time-shifting. With the given F(s) function, one can easily calculate the time shift.

Thanks again