How to Calculate Power Spectral Density S(jw) of a Signal?

[evol_w10lv]

Homework Statement

First of all, I have to calculate power spectral density S(jw) of this signal:

It looks something like that:

t>=0

Homework Equations

The Attempt at a Solution

It seems to me, that I can't use this standart formula:

So.. I see that there is signal multiplication- exponent and sinus, but still not clear how to get spectral density S(jw).
Any ideas?

 

[LCKurtz]

This problem is definitely not in my comfort zone, but why can't you use$$
\sin(wt) = \frac{e^{iwt}-e^{-iwt}}{2i}$$and use the standard formula with just exponentials?

 

[evol_w10lv]

I have got limit from 0 to infinity. I guess, when I use infinity as limit, then I will get e^(inf) which is infinity and then all solution will be infinity. But speaking about the task.. in my opinion there have to use other method, but I don't know which.

 

[LCKurtz]

But your $S(t)$ has $e^{-at}$ in it. If $a$ is large enough, won't that help you at your $\infty$ limits? I haven't worked it out because that's your job. Have you actually tried it? And your graphic shows only for $t>0$.

 

[evol_w10lv]

Also I noticed, that we can't use the same "w" in exponent and sinus.
So, when I use this formula, then I get:

But what about range? It's from 0 to infinity, but I still don't understand some maths there. I guess, that when limit is 0, than all equation is 0. But what about infinity?
-U*exp(-t*(a+jw)) = -U [when t=inf]
What is cos(t*w0) and sin(t*w0), when t=inf?

 

[LCKurtz]

Also I noticed, that we can't use the same "w" in exponent and sinus.
So, when I use this formula, then I get:

But what about range? It's from 0 to infinity, but I still don't understand some maths there. I guess, that when limit is 0, than all equation is 0.

Why do you guess everything is zero? Did you try it?

But what about infinity?
-U*exp(-t*(a+jw)) = -U [when t=inf]
What is cos(t*w0) and sin(t*w0), when t=inf?

What happens to terms like $e^{-at}\sin(\omega_0 t)$ and terms like $e^{-at}e^{iwt}$ when $t\to\infty$?

 

[evol_w10lv]

It's not 0, I tried it, but I found out, that we can't just use standart formula, because a sine wave spectral density has only a delta function at the carrier frequency since the signal contains just
one spectral component namely the carrier frequency.
So I need to do this task in other way. I think using this property:

When I try it with standart formula:

But it don't give same signal as it was at the beginning, when I use inverse formula:

 

[rude man]

1. What is the Fourier integral X(f) of this time function x(t)?
2. What is the energy represented in x(t)? Hint: invoke Parseval's theorem.
3. What is the formula for power, given X(f)? Recall that power = energy averaged over time.
4. What then is the power spectrum G(f)? 2G(f), when integrated over all frequencies, gives you the power.

Note that I'm using only positive frequencies, that's why it's 2G(f) instead of G(f). That factr also impacts your computation of time-averaged energy.

 

[LCKurtz]

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I will let Rude man take it from here because he obviously knows about this stuff.

But do you not see that the "standard formula" you mention above is the answer you get when you work out the integral you started with?

 

[evol_w10lv]

But do you not see that the "standard formula" you mention above is the answer you get when you work out the integral you started with?

Are you saying, that it should be the correct answer?

And here is the way, how I get it, if it's necessary:

But, when I use this formula:

(inverse formula of spectral density or inverse Fourier trasnsform formula)
I should get back s(t) as it was at the beginng, but it's not. Or maybe I can't use inverse formula in this case?
If we are speaking about requirement for the integral to converge, I'm not convinced about this method, because I'm not sure wheather this 'infinity' criterie is acceptable or not when we got sinus. I knpw that pure sine in the time domain evaluates to a delta function in the frequency domain, but in my case it is combinated with exponential term. It should leads to a spread of the energy in the frequency domain, as one conversant person said, but still I'm confused.

1. What is the Fourier integral X(f) of this time function x(t)?
2. What is the energy represented in x(t)? Hint: invoke Parseval's theorem.
3. What is the formula for power, given X(f)? Recall that power = energy averaged over time.
4. What then is the power spectrum G(f)? 2G(f), when integrated over all frequencies, gives you the power.

1. If it's angular freqvency, than formula is almost the same as PSD forumula:

And then result shuld be the same:
Result without limits:

Result with limits (0..inf):

But same understanting, which I mentioned to LCKurtz.
Maybe I have to use two Fourier function multiplication?
Can you explain something more to clear my doubts?
Then I'll continue the task, considering your points. And at the end, as I understand, after I use Parseval's theorem, to get power spectral density, I only need to square the modulus?

 

[rude man]

1. If it's angular freqvency, than formula is almost the same as PSD forumula:

PSD of x(t) is not simply the Fourier transform of x(t). Power immediately implies some kind of squared function. (For example, power dissipated in a 1 ohm resistor = V², V is volts). So my hint is to consider squaring X(f), which is energy (Parseval), then finding the total energy divided by the total time which is the average power.

PSD integrated from f = -∞ to +∞ is the average power in the signal. So two times the integral of PSD over the positive frequency range of interest is also the average power in the signal. That should get you the correct expression for PSD.

P.S. I like to use f instead of ω.
So my Fourier transform is X(f) = ∫x(t)exp(-jwt)dt with ω = 2πf and working with f from then on.

 

[evol_w10lv]

I think, that we are speaking about different things. My fault.
And I don't have to find PSD, if it's:

And I don't have to find energy spectral density neither:

It was non-understanding with language and translation..
So.. I still don't know, how exactly I should say, but it's like Fourier transform. I have to calculate function in frequency domain.
For example:

I have to calculate my task like in this example.
And after that, also there are few extra tasks linked with S(jw), but now.. it means that it is Fourier transform and this is the answer of my s(t)?

 

[LCKurtz]

You never responded to my questions, quoted below, in post #4.

What happens to terms like $e^{-at}\sin(\omega_0 t)$ and terms like $e^{-at}e^{iwt}$ when $t\to\infty$?

and this is the answer of my s(t)?

So I don't know if you understand or not why that is the answer.

 

[evol_w10lv]

But your $S(t)$ has $e^{-at}$ in it. If $a$ is large enough, won't that help you at your $\infty$ limits? I haven't worked it out because that's your job. Have you actually tried it? And your graphic shows only for $t>0$.

In that case it gives 0 and then all part f(inf) gives 0.
There was doubts in my mind about Dirichlet's condition and sin(inf*w), but of course.. e^(-a*inf)=0, then it not matters.
Now it's clear. Thanks.

 

[rude man]

OK, if all you need is the Fourier transform of your time function then follow the advice in posts 2 and 4.
The Fourier transform definitely exists.

 

[evol_w10lv]

Also I have to calculate amplitude spectrum from S(jw).

But it's quite complicated s(jw) to separate real and imaginary part.

Are there is no easiest method?
So here is easy example:

What's the way in my S(jw) case? Must use this method?

 

[rude man]

I agree, the math is messy.

You should also consider that mutliplication in the time domain = convolution in the frequency domain.

 

[evol_w10lv]

Task is done. Thanks.