Phase and amplitude spectrum of signal

[etf]

Hi!

1. Homework Statement

My task is to calculate amplitude and phase spectrum of this signal:

Homework Equations

My idea is to calculate complex Fourier series of this signal, $$f(t)=\sum_{n=-\infty}^{n=+\infty}Fne^{j\frac{2n\pi t}{T}},$$ where $$Fn=\frac{1}{T}\int_{0}^{T}f(t)e^{-j\frac{2n\pi t}{T}}. $$Fn will be some complex number, which can be written as $$|Fn|e^{j\Theta n},$$ where $$|Fn|$$ is amplitude spectrum and $$\Theta n$$ is phase spectrum.

The Attempt at a Solution

I got $$Fn=\frac{E\tau }{T}\frac{\sin{(nw0\tau /2)}}{nw0\tau /2}e^{-jnw0(t1+\tau /2)}, $$ where w0=2*pi/T. We see that phase spectrum is $$\Theta n=-nw0(t1+\tau /2)$$ and amplitude spectrum is $$|Fn|=\frac{E\tau }{T}\frac{\sin{(nw0\tau /2)}}{nw0\tau /2}.$$ Now for some values of $$\tau, $$ $$E$$ and $$T$$ I can plot amplitude and phase spectrum as function of n?

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[collinsmark]

Yeah, that looks right to me. :)

The assumption of course is that f(t) is periodic, and that same period repeats itself from -infinity to infinity. (Otherwise you need to use the Fourier transform instead.)

By the way, notation wise, you forgot your dt when setting up your integral, and I'm also assuming that several times when you wrote Fn you actually meant F_n. Other than stuff like that, it looks good to me.

 

[lazyaditya]

What about the value at n=0 ?

 

[lazyaditya]

There must be a dc term too.

 

[collinsmark]

What about the value at n=0 ?

There must be a dc term too.

Yes, that's right. Perhaps one could try to evaluate the limit of \frac{\sin x}{x} as x \rightarrow 0.

(Hint: l'Hôpital to the rescue)

[Edit: btw, lazyaditya, recall that n goes from -\infty to \infty. So n = 0 is in the middle there somewhere. :)]

 

[lazyaditya]

L'hospital rule can't be applied to discrete sequence and since Fourier series of a periodic signal is discrete in nature thus dc term need to be calculated by keeping n=0 in the equation used for calculating Fourier series coefficient .

 

[collinsmark]

L'hospital rule can't be applied to discrete sequence and since Fourier series of a periodic signal is discrete in nature thus dc term need to be calculated by keeping n=0 in the equation used for calculating Fourier series coefficient .

Correct, if you want to be mathematically rigorous about it, one cannot technically use l'Hopital's rule for a discrete value, as you say. This thread was posted in the engineering section though. We're not the most rigorous bunch.

[Edit, But yes, lazyaditya's method is preferred; it is mathematically better and less likely to get you into trouble in future problems. Treat n = 0 differently than all the other ns, if you would otherwise find yourself in a divide by 0 situation. Evaluate the integral substituting 0 for n in the original integral to produce F_0 specifically. It turns out in this case the answer you get is the same either way (such as using the non-rigorous \frac{\sin x}{x} =1, when x approaches 0) in this particular problem, but the rigorous approach is better in general.]

 

[etf]

@collinsmark
Yes, I forgot dt and I didn't know how to write "n" in index :)
@https://www.physicsforums.com/members/lazyaditya.394087/
I didn't notice that for n=0, Fn is undefined. With substitution n=0 in Fn I got F0=E*tau/T.

 

[etf]

And that's general rule when Fn is undefined for some value n, I put n in original formula and calculate?

 

[etf]

One more question: If I have $$|Fn|$$ and $$\Theta n$$ of some signal $$f(t),$$ and I want to calculate phase and amplitude spectrum of $$f(t+t1),$$ where t1 is some time shift, can I use phase and amplitude spectrum which I have already found for $$f(t)$$ to calculate phase and amplitude shift for $$f(t+t1)$$, or I have to start with calculation from beginning (Represent f(t+t1) in terms of complex Fourier series, calculate |Fn| etc)?

 

[lazyaditya]

Yes, you can use it.

 

[etf]

So if $$f(t)=\sum_{n=-\infty }^{n=\infty }Fne^{jnw0t}, $$ $$f(t+t1)$$ will be $$\sum_{n=-\infty }^{n=\infty }Fne^{jnw0(t+t1)}$$ and it will have same amplitude and phase spectrum as $$f(t)$$? I'm sorry to bother you but I have exam very soon and I'm trying to learn it...

 

[lazyaditya]

Yes it is correct and you are not bothering anyone .

 

[collinsmark]

One more question: If I have $$|Fn|$$ and $$\Theta n$$ of some signal $$f(t),$$ and I want to calculate phase and amplitude spectrum of $$f(t+t1),$$ where t1 is some time shift, can I use phase and amplitude spectrum which I have already found for $$f(t)$$ to calculate phase and amplitude shift for $$f(t+t1)$$, or I have to start with calculation from beginning (Represent f(t+t1) in terms of complex Fourier series, calculate |Fn| etc)?

Changing f(t) \rightarrow f(t + t_1) will cause a difference in the phase response. Specifically, the new result will be the old result multiplied by a complex spiral, sometimes called a corkscrew function, of the form e^{j n \{ \mathrm{something} \} t_1}. In other words, F_n \rightarrow F_n e^{j n \{ \mathrm{something} \} t_1}. I'll let you work out what that something is.

 

[etf]

I think I got it :) I created new thread with some problem which involve time shift...