MHB Calculating Harmonics from FFT of sin(x) Function

[bugatti79]

Hi Folks,

The Fourier Cosine Transform of cos(x) for 0<x<a and 0 everywhere else is given as

F(\omega)=\displaystyle\frac{1}{\sqrt{2 \pi}}[\frac{\sin a (1-\omega)}{1-\omega}+\frac{\sin a (1+\omega)}{1+\omega}]

I can plot this and we get a continuous amlitude spectrum of F(\omega) against (\omega)

but how do I extract/obtain the harmonic of this function which we know has just one harmonic. How do i extract this mathematically and/or from the graph say?

Thanks

 

[chisigma]

Hi Folks,

The Fourier Cosine Transform of cos(x) for 0<x<a and 0 everywhere else is given as

F(\omega)=\displaystyle\frac{1}{\sqrt{2 \pi}}[\frac{\sin a (1-\omega)}{1-\omega}+\frac{\sin a (1+\omega)}{1+\omega}]

I can plot this and we get a continuous amlitude spectrum of F(\omega) against (\omega)

but how do I extract/obtain the harmonic of this function which we know has just one harmonic. How do i extract this mathematically and/or from the graph say?

Thanks

The function would have only one harmonic if it were defined as $\cos x$ for $- \infty< x < + \infty$ ... but it is not so ...

Kind regards

$\chi$ $\sigma$

 

[bugatti79]

The function would have only one harmonic if it were defined as $\cos x$ for $- \infty< x < + \infty$ ... but it is not so ...

Kind regards

$\chi$ $\sigma$

Is there not one harmonic over 1 period?

In either case, is it possible to determine how many harmonics there are? The reason I ask is because i want to find the harmonics of a more complicated function like an amplitude modulation signal when i do the fft

$$A_c[1+A_1\cos(\omega_1 t+\phi_1)]\cos(\omega_c+\phi_c)$$

thanks

 

[chisigma]

Is there not one harmonic over 1 period?

In either case, is it possible to determine how many harmonics there are? The reason I ask is because i want to find the harmonics of a more complicated function like an amplitude modulation signal when i do the fft

$$A_c[1+A_1\cos(\omega_1 t+\phi_1)]\cos(\omega_c+\phi_c)$$

thanks

... there is a little detail... the function is non periodic... and that means that its spectrum doesn't contain 'harmonics' but is a continuos function...

Kind regards

$\chi$ $\sigma$