Why Does the Fourier Transform Show Magnitude Around 3Hz?

[WCMU101]

Hey all.

On Wiki (http://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform) they have some really good pictures explaining the Fourier transform - see the introduction section. The Fourier transform is of an exponentially decaying sinusoid - where the sinusoid (cosine) has a frequency of 3Hz. I do not understand why the Fourier transform has "magnitude" for frequencies surrounding 3Hz? Isn't 3Hz the only frequency present in the function?

Thanks.

Nick.

 

[Polyrhythmic]

If you express some function through a Fourier series, you need sin/cos functions of different frequencies, depending on the original function. Now if you Fourier transform the latter, you basically regain information of those frequencies. That frequency spectrum obviously needs to contain more than just the value of 3Hz.

 

[DragonPetter]

Hey all.

On Wiki (http://en.wikipedia.org/wiki/Fourier_transform#Properties_of_the_Fourier_transform) they have some really good pictures explaining the Fourier transform - see the introduction section. The Fourier transform is of an exponentially decaying sinusoid - where the sinusoid (cosine) has a frequency of 3Hz. I do not understand why the Fourier transform has "magnitude" for frequencies surrounding 3Hz? Isn't 3Hz the only frequency present in the function?

Thanks.

Nick.

Does the sinusoid look like a periodic function? If it is not periodic, or varies from a pure sine wave, then it is made up of more than just 1 frequency component.

Any time-limited function will need infinite bandwidth to be represented in the frequency domain, and you can see this function is not periodic and begins and stops; its only a pulse.