# Fourier transform of a beat

• I
• Salmone

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 ##$\omega_p+\omega_v=\Omega,\space\omega_p-\omega_v=\omega## and ##\Omega\simeq\omega.## I think it is equal to ##\frac{1}{2}(\delta(\omega_p+\omega_v-\omega)+\delta(\omega_p-\omega_v-\omega)+\delta(\omega_v-\omega_p-\omega)+\delta(-\omega_p-\omega_v-\omega))##, is it right? Delta2 ## Answers and Replies What is the Fourier transform of a beat? For example, I want to calculate the Fourier transform of the function ##f(t)=\cos((\omega_p+\omega_v) t)+\cos((\omega_p-\omega_v)t),## where ##$\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?

Salmone
Do you have some reason to think your answer is not correct?
No, I just want to be sure.

What does ##\Omega\simeq\omega## mean?

Means that ##\Omega## has a similar value of ##\omega##, for example: ##\Omega=30Hz## and ##\omega=28Hz##

Delta2
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.

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##.

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