# Spectral density of signal

1. Nov 13, 2013

### evol_w10lv

1. The problem statement, all variables and given/known data
First of all, I have to calculate power spectral density S(jw) of this signal:

It looks something like that:

t>=0

2. Relevant equations

3. 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?

2. Nov 13, 2013

### 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?

3. Nov 13, 2013

### 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.

4. Nov 13, 2013

### 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 $t>0$.

5. Nov 14, 2013

### 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?

6. Nov 14, 2013

### LCKurtz

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

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

7. Nov 14, 2013

### 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 beggining, when I use inverse formula:

8. Nov 14, 2013

### 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.

9. Nov 14, 2013

### LCKurtz

10. Nov 15, 2013

### evol_w10lv

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. 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?

Last edited: Nov 15, 2013
11. Nov 15, 2013

### rude man

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.

12. Nov 15, 2013

### 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)?

13. Nov 15, 2013

### LCKurtz

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

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

14. Nov 15, 2013

### evol_w10lv

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.

15. Nov 15, 2013

### 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.

16. Nov 15, 2013

### evol_w10lv

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

But it's quite complicated s(jw) to seperate 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?

17. Nov 15, 2013

### rude man

I agree, the math is messy.

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

18. Nov 18, 2013