Fourier transform of a beat

In summary: Omega_2## instead of ##\omega## then I guess you are free to use ##\omega## as the variable of the Fourier transform.
  • #1
Salmone
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What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.##

I think it is equal to ##\frac{1}{2}(\delta(\omega_p+\omega_v-\omega)+\delta(\omega_p-\omega_v-\omega)+\delta(\omega_v-\omega_p-\omega)+\delta(-\omega_p-\omega_v-\omega))##, is it right?
 
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  • #2
Salmone said:
What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.##

I think it is equal to ##\frac{1}{2}(\delta(\omega_p+\omega_v-\omega)+\delta(\omega_p-\omega_v-\omega)+\delta(\omega_v-\omega_p-\omega)+\delta(-\omega_p-\omega_v-\omega))##, is it right?
Do you have some reason to think your answer is not correct?
 
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  • #3
vela said:
Do you have some reason to think your answer is not correct?
No, I just want to be sure.
 
  • #4
What does ##\Omega\simeq\omega## mean?
 
  • #5
Means that ##\Omega## has a similar value of ##\omega##, for example: ##\Omega=30Hz## and ##\omega=28Hz##
 
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  • #6
Only slight problem I see is with the chosen symbols, since you chose ##\omega## for the constant ##\omega_p-\omega_v=\omega## you must use another symbol for the variable of the Fourier transform (that is the omega inside the dirac functions). I know we usually say the Fourier transform of ##f(t)## is $$\hat f(\omega)=...$$ but now you have already chosen ##\omega## to denote something else.
 
  • #7
Delta2 said:
Only slight problem I see is with the chosen symbols, since you chose ##\omega## for the constant ##\omega_p-\omega_v=\omega## you must use another symbol for the variable of the Fourier transform (that is the omega inside the dirac functions). I know we usually say the Fourier transform of ##f(t)## is $$\hat f(\omega)=...$$ but now you have already chosen ##\omega## to denote something else.
I can't edit the post no longer, but let's say ##\Omega=\Omega_1## and ##\omega=\omega_p-\omega_v=\Omega_2##.
 
  • #8
Salmone said:
I can't edit the post no longer, but let's say ##\Omega=\Omega_1## and ##\omega=\omega_p-\omega_v=\Omega_2##.
ok fine if you put ##\Omega_2## instead of ##\omega## then I guess you are free to use ##\omega## as the variable of the Fourier transform
 

1. What is a Fourier transform of a beat?

The Fourier transform of a beat is a mathematical tool used to analyze and break down a complex sound wave, such as a beat, into its individual frequency components. It allows us to see the different frequencies that make up the beat and their relative strengths.

2. How is a Fourier transform of a beat calculated?

The Fourier transform of a beat is calculated by taking the complex sound wave and decomposing it into its individual frequency components using a mathematical formula known as the Fourier transform. This involves breaking down the sound wave into its constituent sine and cosine waves of different frequencies.

3. Why is a Fourier transform of a beat useful?

The Fourier transform of a beat is useful because it allows us to analyze and understand the different frequencies present in a complex sound wave. This is especially important in fields such as music and acoustics, where understanding the frequency components of a sound can help in creating and manipulating it.

4. Can a Fourier transform of a beat be used in other areas besides sound analysis?

Yes, the Fourier transform of a beat can be used in various other fields, such as signal processing, image processing, and data analysis. It is a versatile tool that can help in understanding and analyzing any complex wave or signal.

5. Are there any limitations to using a Fourier transform of a beat?

While the Fourier transform of a beat is a powerful tool, it does have some limitations. It assumes that the sound wave is continuous and infinite, which may not always be the case in real-world scenarios. Additionally, it cannot accurately analyze non-stationary signals, such as a beat with changing frequencies.

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