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

In summary: JohnIn summary, the well known function: f(t) = sin(wt) can be expressed in the time domain as: f(t) = sin(wt) where w is a constant. However, if we want to get the highest frequency component in this expression, we need to first convert it to the frequency domain.
  • #1
LM741
130
0
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 weird !
I know w is a constant (defined as the fundamental 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
 
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  • #2
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?
 
  • #3
thanks - but what about the issue about the frequency appearing the time domain expression?
i.e f(t) = sin (wt) where w is frequency?
 
  • #4
[itex]\omega[/itex] usually denotes the angular frequency -- i.e, how many radians the sine wave goes through in one unit of time. If [itex]\omega = 2 \pi[/itex], 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
 
  • #5
LM741 said:
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?
 
  • #6
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 going to write in it.
 
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  • #7
abdo375 said:
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. :rolleyes:
 
  • #8
cepheid said:
That's because the 'w' IS the frequency of the sinusoid. :rolleyes:

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
 
  • #9
chroot said:
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.
 
  • #10
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:

[tex] f(t) = \sin(\omega_0 t) [/tex]

So [itex] \omega_0 [/itex] is the CONSTANT representing the frequency of the sine wave. If I remember right, the Fourier transform is:

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

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).
 
  • #11
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. :biggrin:
 
  • #12
cepheid said:
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:

[tex] f(t) = \sin(\omega_0 t) [/tex]

So [itex] \omega_0 [/itex] is the CONSTANT representing the frequency of the sine wave. If I remember right, the Fourier transform is:

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

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 please reread your post and then read mine, it's exactly what I meant.
 
  • #13
abdo375 said:
cepheid, could you please 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.
 
  • #14
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.
 
  • #15
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
 
  • #16
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.
 
  • #17
thanks .
 

1. What does the equation "Aaah - sin(wt) - time or frequency domain" represent?

The equation represents a sinusoidal wave in either the time or frequency domain, where "Aaah" represents the amplitude of the wave, "sin" represents the sine function, "w" represents the angular frequency, and "t" represents time.

2. How is the time domain related to the frequency domain in this equation?

In this equation, the time domain and frequency domain are two different representations of the same wave. The time domain shows the variation of the wave over time, while the frequency domain shows the different frequencies present in the wave.

3. What is the significance of the amplitude in this equation?

The amplitude in this equation represents the maximum displacement of the wave from its mean or equilibrium position. It is a measure of the strength or intensity of the wave.

4. How does the value of "w" affect the wave in this equation?

The value of "w" represents the rate of change of the wave, also known as the angular frequency. It determines the frequency of the wave, with higher values of "w" resulting in a higher frequency and faster oscillations.

5. Can this equation be used to model real-world phenomena?

Yes, this equation can be used to model various real-world phenomena, such as sound waves, electromagnetic waves, and mechanical vibrations. It is a fundamental tool in studying and understanding wave behavior and is widely used in many fields of science and engineering.

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