# Aaah - sin(wt) - time or frequency domain?

#### LM741

Aaah!!! - sin(wt) - time or frequency domain???!!!!

hi guys

going a bit blank now....

been thinking a bit too much about time and frequency domain to a point where i've confused myself a bit...

The well known function: f(t) = sin(wt)

It is evident that this expression is in the time domain - but how can we get a frequency component, w , in this expression!!!!! really wierd !!
I know w is a constant (defined as the fundemental frequency) but aren't we sort of mixing time and frequency - which i hear is a bad idea!!!

Think about: If i ask what is the highest frequency component in f(t)=sin(200t), the answer would be 200 rad/sec. This is determined by merely looking at the expression in the time!!! But normally to determine the highest frequency component (or any frequency component) of a functino in time - we need to FIRST convert to the frequency domain!!!!!!

Do you guys see my issue here!!!

If anyone can attempt to shed light on the situation, my appreciation would be much like that of an impulse function: unbounded.

Thanks

John

#### berkeman

Mentor
Just think about the transform of a pure sine wave (in the time domain) into the frequency domain. You only get the one impulse at w (well, one at -w also) in the frequency domain. There's only one component in a pure sine wave, so the phrase "highest component in sin(wt)" doesn't really apply, right?

#### LM741

thanks - but what about the issue about the frequency appearing the time domain expression????
i.e f(t) = sin (wt) where w is frequency???

#### chroot

Staff Emeritus
Gold Member
$\omega$ usually denotes the angular frequency -- i.e, how many radians the sine wave goes through in one unit of time. If $\omega = 2 \pi$, then the sine wave goes through one complete cycle in one period of time, so it's frequency is one cycle per unit time.

- Warren

#### berkeman

Mentor
thanks - but what about the issue about the frequency appearing the time domain expression????
i.e f(t) = sin (wt) where w is frequency???
Not much difference from velocity and distance appearing in equations together, is it?

#### abdo375

The problem is with the way your looking at it, your looking at the "w" in sin(wt) as it's frequency while it's just a constant multiplied by t, which happens to be the same constant at which the delta is shifted when you get the fourier transform of sin(wt).

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berkeman you got a blog, can't wait to see what your gonna write in it.

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#### cepheid

Staff Emeritus
Gold Member
The problem is with the way your looking at it, your looking at the "w" in sin(wt) as it's frequency
That's because the 'w' IS the frequency of the sinusoid. #### chroot

Staff Emeritus
Gold Member
That's because the 'w' IS the frequency of the sinusoid. I think he's trying to say that w is NOT the frequency in terms of cycles per second, it's the frequency in terms of radians per second.

- Warren

#### cepheid

Staff Emeritus
Gold Member
I think he's trying to say that w is NOT the frequency in terms of cycles per second, it's the frequency in terms of radians per second.

- Warren
I assumed that the OP knew as much in the first place. I don't understand how that is relevant to what he was confused about.

#### cepheid

Staff Emeritus
Gold Member
LM741. You shouldn't be bothered by the fact that the (*one and only*) frequency of a signal that varies sinusoidally with time appears in the expression for that signal. I think if you think about it, you'll see that sin(wt) is a signal with (angular) frequency w, where w is a *constant*. So what is the problem with sin(wt) in this context? Bad notation! The letter omega is getting double usage here as a constant representing the (single) frequency of the sine wave and as a variable when we flip to the frequency domain and starting thinking about the signal as a *function* of frequency instead of time. Typically when we're doing Fourier analysis we make this distinction much more explicit:

$$f(t) = \sin(\omega_0 t)$$

So $\omega_0$ is the CONSTANT representing the frequency of the sine wave. If I remember right, the fourier transform is:

$$\mathcal{F}[f(t)] = \frac{\pi}{i}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)]$$

But don't quote me on that =p It was from memory. Anyway, you can see that there is no issue here. In the time domain we have the signal as a function of only one variable (t), and in the frequency domain it is a function of only one variable (omega).

#### Swapnil

cepheid said:
The letter omega is getting double usage here as a constant representing the (single) frequency of the sine wave and as a variable when we flip to the frequency domain and starting thinking about the signal as a *function* of frequency instead of time ... In the time domain we have the signal as a function of only one variable (t), and in the frequency domain it is a function of only one variable (omega).
OHH! Man! I had been confused about this for months. I get it now. Thanks cepheid. #### abdo375

LM741. You shouldn't be bothered by the fact that the (*one and only*) frequency of a signal that varies sinusoidally with time appears in the expression for that signal. I think if you think about it, you'll see that sin(wt) is a signal with (angular) frequency w, where w is a *constant*. So what is the problem with sin(wt) in this context? Bad notation! The letter omega is getting double usage here as a constant representing the (single) frequency of the sine wave and as a variable when we flip to the frequency domain and starting thinking about the signal as a *function* of frequency instead of time. Typically when we're doing Fourier analysis we make this distinction much more explicit:

$$f(t) = \sin(\omega_0 t)$$

So $\omega_0$ is the CONSTANT representing the frequency of the sine wave. If I remember right, the fourier transform is:

$$\mathcal{F}[f(t)] = \frac{\pi}{i}[\delta(\omega - \omega_0) - \delta(\omega + \omega_0)]$$

But don't quote me on that =p It was from memory. Anyway, you can see that there is no issue here. In the time domain we have the signal as a function of only one variable (t), and in the frequency domain it is a function of only one variable (omega).

cepheid, could you plz reread your post and then read mine, it's exactly what I meant.

#### cepheid

Staff Emeritus
Gold Member
cepheid, could you plz reread your post and then read mine, it's exactly what I meant.
Yes,

I can see now that that's what you meant, but given the word "it's", that you used, it didn't seem quite clear (I was confused about what you meant). So I opted to clarify. I didn't mean to step on your toes, and I apologize for my sarcastic reply. It was based on a misinterpretation of what you were trying to say.

#### abdo375

No problem, I guess I have to improve my English to sound less aggressive (in my second post), and more explanatory (in my first post), I seem to project a wrong image with my posts.

#### LM741

thanks guys for all your feedback.
thanks cepheid - sorry about notation, but i was aware that it was a fundemantal angular frequency(i.e. a constant) - i just don't like the idea of it being called a frequency (even though i know it is) when we are in the time domain...but don't worry...ill let it go... thanks

what about sin(200t): whenever i get a functino like this, can i ALWAYS assume that the constant is my angular frequency???? i .e 200 = (2*pi)/T.

Don't some textbooks use radians per second (angular frequency) and some just use seconds??? maybe that 200 has already been divided by 2*pi, therefore its in seconds??? how can i possibly asscertain this??

thank

#### BNoelCMU

I believe the convention is that w is in radians per second. Of course, if you're going to play with this on your calculator, you must make sure you're using the right units (radians or degrees). But to answer your question, yes the constant is always the angular frequency.

thanks .

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