How Do Laplace Transforms Help Understand Frequency and Amplitude in Signals?

[Plecto]

Hi. We are learning about Laplace transforms at uni and I must say that this is a real pain. I have one questions concerning the concept of Laplace transforms, and also a question concerning a specific transform. The task is to make a Laplace transform of: t*sin(2t). I could do an integration by parts and solve it using the standard definition of the Laplace transform, but I don't think that is the idea. The question says "L[f(t)]=F(s). L[t*f(t)]=-dF(s)/ds". I can't find this anywhere in the book other than above the assignment I'm asked to do so there's no explanation of it, I have no idea of what it means :( I was thinking that it might have to do with that L(f')=s*L(f)-f(0), but to use that, I would have to know the Laplace transform of f'(t), but I don't :( Is there anyone that could give me some help?

I'm also struggling to see what the frequency or s-domain actually tells me. Our lecturer gave an example where three sine waves were on top of each other and that it would be difficult to see exactly how many and at what frequencies they were. By doing a Laplace transform we could see the frequency along the s-axis and their amplitude along the y-axis, but what about doing the Laplace transform of a constant? A constant doesn't have a frequency, neither does a function like e^t. The Laplace transform of e^t is 1/s, what kind of information will that function give me? I can set s=2 which will give y=0.5, does it say that the amplitude is 0.5 when the frequency is 2hz? That makes no sense at all :(

 

[UltrafastPED]

A good tutorial will help you get started: http://tutorial.math.lamar.edu/Classes/DE/LaplaceIntro.aspx

In your particular problem they have given you a special "product formula", for L[t*f(t)], where f(t) is any function such that L[f(t)] = F(s) ... so it is completely general.

The L[t*f(t)] = -d/ds[F(s)] ... you just take the derivative of F(s) times minus one.

In general, the Laplace transform is a member of a family of "integral transforms"; they create a 1-1 mapping from the domain where you started (time domain here) to a new domain (frequency domain) - and they are used because the problems are easier to solve in the new domain; then you transform back to the original domain.

In many cases there is a physical interpretation of the transform, and when you become familiar with it you can analyze problems in the s-domain without needing to go back and forth. The Fourier transform is a special case of the Laplace transform, and with just a bit of exposure people are "happy" to think in the frequency domain. The same happens in the s-domain, but perhaps it takes a bit longer!

This set of notes almost derives the formula which you have; you can easily finish the derivation:
http://www.stanford.edu/~boyd/ee102/laplace.pdf

 

[Plecto]

Alright, thanks for the response :) I had math at high school (equivalent) in 2008, I then took first year uni math in 2010 (barely made it), and now I'm taking second year uni math so there's a lot of holes in my knowledge :( I'm struggling to remember simple integration, differentiation etc.

So I take the Laplace transform of sin(2t) which gives me 2/(s^2+4), I then take the derivative of that and multiply it by -1 which gives me 4s/(s^2+4)^2 which is the right answer :) I don't get the right answer when trying to do the laplace transform of t*cos(3t) though. The Laplace transform of cos(3t) is s/(s^2+9), taking the derivative and multiplying by -1 gives me (-9-s^2)/(s^2+9)^2. The correct answer should be (s^2-9)/(s^2+9)^2 so I'm not sure what I'm doing wrong :(

I'll take a further look at the tutorial you posted. I've also watched a bunch of videos on youtube, but those video's doesn't seem to address everything.

The next question states that L[f(t)]/t=∫(from s to ∞) F(u). I think it's tiresome that these statements isn't mentioned anywhere else, no explanation or examples at all :( I don't know what F(u) is, I haven't worked with 'u' as a variable anywhere :( What function is F(u) exactly?

 

[UltrafastPED]

The Laplace transform of cos(3t) is s/(s^2+9), taking the derivative and multiplying by -1 gives me (-9-s^2)/(s^2+9)^2. The correct answer should be (s^2-9)/(s^2+9)^2 so I'm not sure what I'm doing wrong :(

The derivative should give 1/(s^2+9) - s(2s)/(s^2+9) = (9-s^2)/(s^2+9) ... which then gives the correct answer when multiplied by -1 for the t factor. So double-check your derivative to see where it went wrong.

The next question states that L[f(t)]/t=∫(from s to ∞) F(u). I think it's tiresome that these statements isn't mentioned anywhere else, no explanation or examples at all :( I don't know what F(u) is, I haven't worked with 'u' as a variable anywhere :( What function is F(u) exactly?

u(t) is the unit step function; =0 for t<0, =1 for t>0. It is the function used to model a switch in a circuit. But I would need more context to be sure.